Explanation:
Okay, so I need to find the degree measure of the least positive angle θ such that tanθ equals this big fraction. Let me first write down the given expression to make it clearer:

Numerator: cos5°cos20° + cos35°cos50° - sin5°sin20° - sin35°sin50°

Denominator: sin5°cos20° - sin35°cos50° + cos5°sin20° - cos35°sin50°

Hmm, that looks complicated, but maybe I can simplify both numerator and denominator using trigonometric identities. Let me recall some identities.

First, I remember that cosA cosB - sinA sinB = cos(A + B). Similarly, sinA cosB + cosA sinB = sin(A + B). Wait, let me check:

Yes, cos(A + B) = cosA cosB - sinA sinB, and sin(A + B) = sinA cosB + cosA sinB. So maybe I can group terms in numerator and denominator accordingly.

Looking at the numerator:

cos5°cos20° - sin5°sin20° + cos35°cos50° - sin35°sin50°

Wait, that's two terms of the form cosA cosB - sinA sinB. So that would be cos(5° + 20°) + cos(35° + 50°). Let me compute those angles:

5° + 20° = 25°, and 35° + 50° = 85°. So numerator simplifies to cos25° + cos85°.

Wait, let me check again. The original numerator is cos5°cos20° + cos35°cos50° - sin5°sin20° - sin35°sin50°. So grouping the first two terms and the last two terms:

[cos5°cos20° - sin5°sin20°] + [cos35°cos50° - sin35°sin50°] = cos(5° + 20°) + cos(35° + 50°) = cos25° + cos85°. Yes, that seems right.

Now the denominator: sin5°cos20° - sin35°cos50° + cos5°sin20° - cos35°sin50°

Let me rearrange the terms to see if I can group them. Let me write them as:

[sin5°cos20° + cos5°sin20°] - [sin35°cos50° + cos35°sin50°]

Because the first two terms are positive and the last two are negative. Let me check:

Original denominator: sin5°cos20° - sin35°cos50° + cos5°sin20° - cos35°sin50°

Grouping first and third terms: sin5°cos20° + cos5°sin20°, and then - sin35°cos50° - cos35°sin50°. So that's:

[sin5°cos20° + cos5°sin20°] - [sin35°cos50° + cos35°sin50°]

Which is sin(5° + 20°) - sin(35° + 50°) because sin(A + B) = sinA cosB + cosA sinB. Therefore:

sin25° - sin85°.

So now, the original expression simplifies to:

Numerator: cos25° + cos85°

Denominator: sin25° - sin85°

So tanθ = [cos25° + cos85°] / [sin25° - sin85°]

Now, I need to simplify this expression. Let me recall sum-to-product formulas. For cosA + cosB, it's 2 cos[(A+B)/2] cos[(A-B)/2], and for sinA - sinB, it's 2 cos[(A+B)/2] sin[(A-B)/2]. Let me apply these.

First, numerator: cos25° + cos85°. Let me set A = 25°, B = 85°. Then:

cosA + cosB = 2 cos[(A+B)/2] cos[(A - B)/2] = 2 cos[(25 + 85)/2] cos[(25 - 85)/2] = 2 cos(110°/2) cos(-60°/2) = 2 cos55° cos(-30°). Since cosine is even, cos(-30°) = cos30°, so numerator becomes 2 cos55° cos30°.

Similarly, denominator: sin25° - sin85°. Let me use the identity sinA - sinB = 2 cos[(A+B)/2] sin[(A - B)/2]. Let me set A = 25°, B = 85°, so:

sin25° - sin85° = 2 cos[(25 + 85)/2] sin[(25 - 85)/2] = 2 cos(110°/2) sin(-60°/2) = 2 cos55° sin(-30°). Since sin(-x) = -sinx, this becomes 2 cos55° (-sin30°) = -2 cos55° sin30°.

Therefore, tanθ = [2 cos55° cos30°] / [-2 cos55° sin30°].

Wait, the 2 cos55° cancels out from numerator and denominator, so tanθ = [cos30°] / [-sin30°] = -cot30°. Because cos30° / sin30° is cot30°, so with the negative sign, it's -cot30°.

But cot30° is 1/tan30°, which is √3. Therefore, tanθ = -√3.

So tanθ = -√3. Now, we need to find the least positive angle θ such that tanθ = -√3.

But tanθ is negative, so θ is in a quadrant where tangent is negative, which is second or fourth quadrant. The principal value of arctangent of -√3 is -60°, but since we need the least positive angle, we can add 180° to -60° to get 120°, or add 360° to get 300°, but 120° is smaller.

But let me check. The tangent function has period 180°, so solutions are θ = -60° + 180°k, where k is integer. The least positive angle would be when k=1: -60° + 180° = 120°, and k=0 gives -60°, which is negative. So 120° is the least positive angle.

But let me confirm if there's a smaller positive angle. For example, if we consider θ in the fourth quadrant, tanθ = -√3 would be 360° - 60° = 300°, which is larger than 120°, so 120° is indeed the smallest positive angle.

But wait, let me check if my simplification steps were correct. Let me go back through the steps again to make sure I didn't make a mistake.

Starting with numerator: cos25° + cos85°. Using sum to product: 2 cos[(25+85)/2] cos[(25-85)/2] = 2 cos(55°) cos(-30°). Since cos(-30°) = cos30°, so 2 cos55° cos30°. Correct.

Denominator: sin25° - sin85° = 2 cos[(25+85)/2] sin[(25-85)/2] = 2 cos55° sin(-30°) = -2 cos55° sin30°. Correct. Then tanθ = [2 cos55° cos30°]/[-2 cos55° sin30°] = -cot30°. Yes, because cos30° / sin30° is cot30°, so negative of that is -cot30°.

cot30° is 1/tan30°, which is √3. So tanθ = -√3. Correct.

So θ is arctangent of -√3. As mentioned, arctangent of √3 is 60°, so arctangent of -√3 is -60°, but adding 180° gives 120°, which is in the second quadrant where tangent is negative. So θ = 120°.

But let me check if there's another way to think about this. Maybe there's a mistake in the sign? Let me check the denominator again.

Original denominator: sin5°cos20° - sin35°cos50° + cos5°sin20° - cos35°sin50°.

Grouped as [sin5°cos20° + cos5°sin20°] - [sin35°cos50° + cos35°sin50°] = sin(5° + 20°) - sin(35° + 50°) = sin25° - sin85°. Correct.

Then sin25° - sin85°. Let me compute sin25° - sin85° numerically to check. Let me recall that sin85° is sin(90° - 5°) = cos5° ≈ 0.9962. sin25° ≈ 0.4226. So 0.4226 - 0.9962 ≈ -0.5736.

Similarly, numerator: cos25° + cos85°. cos25° ≈ 0.9063, cos85° ≈ 0.0872. Sum ≈ 0.9063 + 0.0872 ≈ 0.9935.

So tanθ ≈ 0.9935 / (-0.5736) ≈ -1.732. Which is approximately -√3 ≈ -1.732. So that checks out.

Therefore, tanθ = -√3, so θ is 120°, since tan(120°) = tan(180° - 60°) = -tan60° = -√3. Correct.

But let me check if there's a smaller positive angle. For example, if θ is in the fourth quadrant, 360° - 60° = 300°, which is larger than 120°, so 120° is indeed the smallest positive angle.

Wait, but maybe there's another angle? Let me think. The period of tangent is 180°, so the solutions are θ = -60° + 180°k. So for k=1, θ=120°, for k=2, θ=300°, etc. So yes, 120° is the smallest positive.

Therefore, the answer is 120°, so θ = 120°, so the degree measure is 120.

But let me check once again if I didn't make a mistake in simplifying numerator and denominator. Let me re-express the original numerator and denominator again.

Original numerator: cos5°cos20° + cos35°cos50° - sin5°sin20° - sin35°sin50°.

Grouped as [cos5°cos20° - sin5°sin20°] + [cos35°cos50° - sin35°sin50°] = cos(5° + 20°) + cos(35° + 50°) = cos25° + cos85°. Correct.

Original denominator: sin5°cos20° - sin35°cos50° + cos5°sin20° - cos35°sin50°.

Grouped as [sin5°cos20° + cos5°sin20°] - [sin35°cos50° + cos35°sin50°] = sin(5° + 20°) - sin(35° + 50°) = sin25° - sin85°. Correct.

So tanθ = [cos25° + cos85°]/[sin25° - sin85°] = [2 cos55° cos30°]/[-2 cos55° sin30°] = -cot30° = -√3. Correct.

Therefore, θ = arctan(-√3). Since we need the least positive angle, θ = 180° - 60° = 120°.

Alternatively, maybe there's another approach? Let me think. For example, maybe using complex numbers or other identities? But I think the steps are correct.

Alternatively, maybe I can compute the numerator and denominator numerically and then compute tanθ? Let me try that as a check.

Compute numerator:

cos5° ≈ 0.9962, cos20° ≈ 0.9397, so cos5°cos20° ≈ 0.9962 * 0.9397 ≈ 0.935.

cos35° ≈ 0.8192, cos50° ≈ 0.6428, so cos35°cos50° ≈ 0.8192 * 0.6428 ≈ 0.527.

sin5° ≈ 0.0872, sin20° ≈ 0.3420, so sin5°sin20° ≈ 0.0872 * 0.3420 ≈ 0.0298.

sin35° ≈ 0.5736, sin50° ≈ 0.7660, so sin35°sin50° ≈ 0.5736 * 0.7660 ≈ 0.439.

Therefore, numerator ≈ 0.935 + 0.527 - 0.0298 - 0.439 ≈ (0.935 + 0.527) - (0.0298 + 0.439) ≈ 1.462 - 0.4688 ≈ 0.9932. Which matches the earlier approximate value of 0.9935.

Denominator:

sin5°cos20° ≈ 0.0872 * 0.9397 ≈ 0.0819.

-sin35°cos50° ≈ -0.5736 * 0.6428 ≈ -0.368.

cos5°sin20° ≈ 0.9962 * 0.3420 ≈ 0.3406.

-cos35°sin50° ≈ -0.8192 * 0.7660 ≈ -0.628.

Adding them up: 0.0819 - 0.368 + 0.3406 - 0.628 ≈ (0.0819 + 0.3406) - (0.368 + 0.628) ≈ 0.4225 - 0.996 ≈ -0.5735. Which matches the earlier approximate value of -0.5736.

Therefore, tanθ ≈ 0.9932 / (-0.5735) ≈ -1.732, which is indeed -√3 ≈ -1.732. So that's correct.

Therefore, θ is 120°, as previously determined.

But just to be thorough, let me check tan(120°). tan(120°) = tan(180° - 60°) = -tan(60°) = -√3. Correct.

Alternatively, if I think about the unit circle, 120° is in the second quadrant, where tangent is negative, and the reference angle is 60°, so tan(120°) = -tan(60°) = -√3.

Therefore, θ = 120° is indeed the correct answer.

I think that's solid. I don't see any mistakes in my reasoning. So the least positive angle θ is 120 degrees.

**Final Answer**
\boxed{120}

To solve the given problem, we are asked to find the **least positive angle** $ \theta $ such that:

$$
\tan \theta = \frac{\cos 5^\circ \cos 20^\circ + \cos 35^\circ \cos 50^\circ - \sin 5^\circ \sin 20^\circ - \sin 35^\circ \sin 50^\circ}{\sin 5^\circ \cos 20^\circ - \sin 35^\circ \cos 50^\circ + \cos 5^\circ \sin 20^\circ - \cos 35^\circ \sin 50^\circ}
$$

---

### Step 1: Simplify the Numerator

We observe that the numerator can be grouped as:

$$
(\cos 5^\circ \cos 20^\circ - \sin 5^\circ \sin 20^\circ) + (\cos 35^\circ \cos 50^\circ - \sin 35^\circ \sin 50^\circ)
$$

Using the identity:

$$
\cos A \cos B - \sin A \sin B = \cos(A + B)
$$

We get:

$$
\cos(5^\circ + 20^\circ) + \cos(35^\circ + 50^\circ) = \cos 25^\circ + \cos 85^\circ
$$

---

### Step 2: Simplify the Denominator

The denominator is:

$$
(\sin 5^\circ \cos 20^\circ + \cos 5^\circ \sin 20^\circ) - (\sin 35^\circ \cos 50^\circ + \cos 35^\circ \sin 50^\circ)
$$

Using the identity:

$$
\sin A \cos B + \cos A \sin B = \sin(A + B)
$$

We get:

$$
\sin(5^\circ + 20^\circ) - \sin(35^\circ + 50^\circ) = \sin 25^\circ - \sin 85^\circ
$$

---

### Step 3: Apply Sum-to-Product Identities

Now we simplify both numerator and denominator using sum-to-product identities:

- **Numerator:**
  $$
  \cos 25^\circ + \cos 85^\circ = 2 \cos\left(\frac{25^\circ + 85^\circ}{2}\right) \cos\left(\frac{25^\circ - 85^\circ}{2}\right) = 2 \cos 55^\circ \cos(-30^\circ)
  $$
  Since $ \cos(-x) = \cos x $, this becomes:
  $$
  2 \cos 55^\circ \cos 30^\circ
  $$

- **Denominator:**
  $$
  \sin 25^\circ - \sin 85^\circ = 2 \cos\left(\frac{25^\circ + 85^\circ}{2}\right) \sin\left(\frac{25^\circ - 85^\circ}{2}\right) = 2 \cos 55^\circ \sin(-30^\circ)
  $$
  Since $ \sin(-x) = -\sin x $, this becomes:
  $$
  -2 \cos 55^\circ \sin 30^\circ
  $$

---

### Step 4: Simplify the Expression

Now, the expression becomes:

$$
\tan \theta = \frac{2 \cos 55^\circ \cos 30^\circ}{-2 \cos 55^\circ \sin 30^\circ}
$$

Cancel $ 2 \cos 55^\circ $ from numerator and denominator:

$$
\tan \theta = \frac{\cos 30^\circ}{-\sin 30^\circ} = -\cot 30^\circ
$$

Recall that $ \cot 30^\circ = \frac{1}{\tan 30^\circ} = \sqrt{3} $, so:

$$
\tan \theta = -\sqrt{3}
$$

---

### Step 5: Find the Least Positive Angle

We are to find the **least positive angle** $ \theta $ such that $ \tan \theta = -\sqrt{3} $.

The tangent function is negative in the **second and fourth quadrants**. The reference angle for $ \tan^{-1}(\sqrt{3}) = 60^\circ $, so:

- In the **second quadrant**: $ \theta = 180^\circ - 60^\circ = 120^\circ $
- In the **fourth quadrant**: $ \theta = 360^\circ - 60^\circ = 300^\circ $

Since we are looking for the **least positive** angle, the answer is:

$$
\boxed{120}
$$

[Assistant]
Explanation:
Okay, so I need to find the degree measure of the least positive angle θ such that tanθ equals this big fraction. Let me first write down the given expression to make it clearer:

Numerator: cos5°cos20° + cos35°cos50° - sin5°sin20° - sin35°sin50°

Denominator: sin5°cos20° - sin35°cos50° + cos5°sin20° - cos35°sin50°

Hmm, that looks complicated, but maybe I can simplify both numerator and denominator using trigonometric identities. Let me recall some identities.

First, I remember that cosA cosB - sinA sinB = cos(A + B). Similarly, sinA cosB + cosA sinB = sin(A + B). Wait, let me check:

Yes, cos(A + B) = cosA cosB - sinA sinB, and sin(A + B) = sinA cosB + cosA sinB. So maybe I can group terms in numerator and denominator accordingly.

Looking at the numerator:

cos5°cos20° - sin5°sin20° + cos35°cos50° - sin35°sin50°

Wait, that's two terms of the form cosA cosB - sinA sinB. So that would be cos(5° + 20°) + cos(35° + 50°). Let me compute those angles:

5° + 20° = 25°, and 35° + 50° = 85°. So numerator simplifies to cos25° + cos85°.

Wait, let me check again. The original numerator is cos5°cos20° + cos35°cos50° - sin5°sin20° - sin35°sin50°. So grouping the first two terms and the last two terms:

[cos5°cos20° - sin5°sin20°] + [cos35°cos50° - sin35°sin50°] = cos(5° + 20°) + cos(35° + 50°) = cos25° + cos85°. Yes, that seems right.

Now the denominator: sin5°cos20° - sin35°cos50° + cos5°sin20° - cos35°sin50°

Let me rearrange the terms to see if I can group them. Let me write them as:

[sin5°cos20° + cos5°sin20°] - [sin35°cos50° + cos35°sin50°]

Because the first two terms are positive and the last two are negative. Let me check:

Original denominator: sin5°cos20° - sin35°cos50° + cos5°sin20° - cos35°sin50°

Grouping first and third terms: sin5°cos20° + cos5°sin20°, and then - sin35°cos50° - cos35°sin50°. So that's:

[sin5°cos20° + cos5°sin20°] - [sin35°cos50° + cos35°sin50°]

Which is sin(5° + 20°) - sin(35° + 50°) because sin(A + B) = sinA cosB + cosA sinB. Therefore:

sin25° - sin85°.

So now, the original expression simplifies to:

Numerator: cos25° + cos85°

Denominator: sin25° - sin85°

So tanθ = [cos25° + cos85°] / [sin25° - sin85°]

Now, I need to simplify this expression. Let me recall sum-to-product formulas. For cosA + cosB, it's 2 cos[(A+B)/2] cos[(A-B)/2], and for sinA - sinB, it's 2 cos[(A+B)/2] sin[(A-B)/2]. Let me apply these.

First, numerator: cos25° + cos85°. Let me set A = 25°, B = 85°. Then:

cosA + cosB = 2 cos[(A+B)/2] cos[(A - B)/2] = 2 cos[(25 + 85)/2] cos[(25 - 85)/2] = 2 cos(110°/2) cos(-60°/2) = 2 cos55° cos(-30°). Since cosine is even, cos(-30°) = cos30°, so numerator becomes 2 cos55° cos30°.

Similarly, denominator: sin25° - sin85°. Let me use the identity sinA - sinB = 2 cos[(A+B)/2] sin[(A - B)/2]. Let me set A = 25°, B = 85°, so:

sin25° - sin85° = 2 cos[(25 + 85)/2] sin[(25 - 85)/2] = 2 cos(110°/2) sin(-60°/2) = 2 cos55° sin(-30°). Since sin(-x) = -sinx, this becomes 2 cos55° (-sin30°) = -2 cos55° sin30°.

Therefore, tanθ = [2 cos55° cos30°] / [-2 cos55° sin30°].

Wait, the 2 cos55° cancels out from numerator and denominator, so tanθ = [cos30°] / [-sin30°] = -cot30°. Because cos30° / sin30° is cot30°, so with the negative sign, it's -cot30°.

But cot30° is 1/tan30°, which is √3. Therefore, tanθ = -√3.

So tanθ = -√3. Now, we need to find the least positive angle θ such that tanθ = -√3.

But tanθ is negative, so θ is in a quadrant where tangent is negative, which is second or fourth quadrant. The principal value of arctangent of -√3 is -60°, but since we need the least positive angle, we can add 180° to -60° to get 120°, or add 360° to get 300°, but 120° is smaller.

But let me check. The tangent function has period 180°, so solutions are θ = -60° + 180°k, where k is integer. The least positive angle would be when k=1: -60° + 180° = 120°, and k=0 gives -60°, which is negative. So 120° is the least positive angle.

But let me confirm if there's a smaller positive angle. For example, if we consider θ in the fourth quadrant, tanθ = -√3 would be 360° - 60° = 300°, which is larger than 120°, so 120° is indeed the smallest positive angle.

But wait, let me check if my simplification steps were correct. Let me go back through the steps again to make sure I didn't make a mistake.

Starting with numerator: cos25° + cos85°. Using sum to product: 2 cos[(25+85)/2] cos[(25-85)/2] = 2 cos(55°) cos(-30°). Since cos(-30°) = cos30°, so 2 cos55° cos30°. Correct.

Denominator: sin25° - sin85° = 2 cos[(25+85)/2] sin[(25-85)/2] = 2 cos55° sin(-30°) = -2 cos55° sin30°. Correct. Then tanθ = [2 cos55° cos30°]/[-2 cos55° sin30°] = -cot30°. Yes, because cos30° / sin30° is cot30°, so negative of that is -cot30°.

cot30° is 1/tan30°, which is √3. So tanθ = -√3. Correct.

So θ is arctangent of -√3. As mentioned, arctangent of √3 is 60°, so arctangent of -√3 is -60°, but adding 180° gives 120°, which is in the second quadrant where tangent is negative. So θ = 120°.

But let me check if there's another way to think about this. Maybe there's a mistake in the sign? Let me check the denominator again.

Original denominator: sin5°cos20° - sin35°cos50° + cos5°sin20° - cos35°sin50°.

Grouped as [sin5°cos20° + cos5°sin20°] - [sin35°cos50° + cos35°sin50°] = sin(5° + 20°) - sin(35° + 50°) = sin25° - sin85°. Correct.

Then sin25° - sin85°. Let me compute sin25° - sin85° numerically to check. Let me recall that sin85° is sin(90° - 5°) = cos5° ≈ 0.9962. sin25° ≈ 0.4226. So 0.4226 - 0.9962 ≈ -0.5736.

Similarly, numerator: cos25° + cos85°. cos25° ≈ 0.9063, cos85° ≈ 0.0872. Sum ≈ 0.9063 + 0.0872 ≈ 0.9935.

So tanθ ≈ 0.9935 / (-0.5736) ≈ -1.732. Which is approximately -√3 ≈ -1.732. So that checks out.

Therefore, tanθ = -√3, so θ is 120°, since tan(120°) = tan(180° - 60°) = -tan60° = -√3. Correct.

But let me check if there's a smaller positive angle. For example, if θ is in the fourth quadrant, 360° - 60° = 300°, which is larger than 120°, so 120° is indeed the smallest positive angle.

Wait, but maybe there's another angle? Let me think. The period of tangent is 180°, so the solutions are θ = -60° + 180°k. So for k=1, θ=120°, for k=2, θ=300°, etc. So yes, 120° is the smallest positive.

Therefore, the answer is 120°, so θ = 120°, so the degree measure is 120.

But let me check once again if I didn't make a mistake in simplifying numerator and denominator. Let me re-express the original numerator and denominator again.

Original numerator: cos5°cos20° + cos35°cos50° - sin5°sin20° - sin35°sin50°.

Grouped as [cos5°cos20° - sin5°sin20°] + [cos35°cos50° - sin35°sin50°] = cos(5° + 20°) + cos(35° + 50°) = cos25° + cos85°. Correct.

Original denominator: sin5°cos20° - sin35°cos50° + cos5°sin20° - cos35°sin50°.

Grouped as [sin5°cos20° + cos5°sin20°] - [sin35°cos50° + cos35°sin50°] = sin(5° + 20°) - sin(35° + 50°) = sin25° - sin85°. Correct.

So tanθ = [cos25° + cos85°]/[sin25° - sin85°] = [2 cos55° cos30°]/[-2 cos55° sin30°] = -cot30° = -√3. Correct.

Therefore, θ = arctan(-√3). Since we need the least positive angle, θ = 180° - 60° = 120°.

Alternatively, maybe there's another approach? Let me think. For example, maybe using complex numbers or other identities? But I think the steps are correct.

Alternatively, maybe I can compute the numerator and denominator numerically and then compute tanθ? Let me try that as a check.

Compute numerator:

cos5° ≈ 0.9962, cos20° ≈ 0.9397, so cos5°cos20° ≈ 0.9962 * 0.9397 ≈ 0.935.

cos35° ≈ 0.8192, cos50° ≈ 0.6428, so cos35°cos50° ≈ 0.8192 * 0.6428 ≈ 0.527.

sin5° ≈ 0.0872, sin20° ≈ 0.3420, so sin5°sin20° ≈ 0.0872 * 0.3420 ≈ 0.0298.

sin35° ≈ 0.5736, sin50° ≈ 0.7660, so sin35°sin50° ≈ 0.5736 * 0.7660 ≈ 0.439.

Therefore, numerator ≈ 0.935 + 0.527 - 0.0298 - 0.439 ≈ (0.935 + 0.527) - (0.0298 + 0.439) ≈ 1.462 - 0.4688 ≈ 0.9932. Which matches the earlier approximate value of 0.9935.

Denominator:

sin5°cos20° ≈ 0.0872 * 0.9397 ≈ 0.0819.

-sin35°cos50° ≈ -0.5736 * 0.6428 ≈ -0.368.

cos5°sin20° ≈ 0.9962 * 0.3420 ≈ 0.3406.

-cos35°sin50° ≈ -0.8192 * 0.7660 ≈ -0.628.

Adding them up: 0.0819 - 0.368 + 0.3406 - 0.628 ≈ (0.0819 + 0.3406) - (0.368 + 0.628) ≈ 0.4225 - 0.996 ≈ -0.5735. Which matches the earlier approximate value of -0.5736.

Therefore, tanθ ≈ 0.9932 / (-0.5735) ≈ -1.732, which is indeed -√3 ≈ -1.732. So that's correct.

Therefore, θ is 120°, as previously determined.

But just to be thorough, let me check tan(120°). tan(120°) = tan(180° - 60°) = -tan(60°) = -√3. Correct.

Alternatively, if I think about the unit circle, 120° is in the second quadrant, where tangent is negative, and the reference angle is 60°, so tan(120°) = -tan(60°) = -√3.

Therefore, θ = 120° is indeed the correct answer.

I think that's solid. I don't see any mistakes in my reasoning. So the least positive angle θ is 120 degrees.

**Final Answer**
\boxed{120}

To solve the given problem, we are asked to find the **least positive angle** $ \theta $ such that:

$$
\tan \theta = \frac{\cos 5^\circ \cos 20^\circ + \cos 35^\circ \cos 50^\circ - \sin 5^\circ \sin 20^\circ - \sin 35^\circ \sin 50^\circ}{\sin 5^\circ \cos 20^\circ - \sin 35^\circ \cos 50^\circ + \cos 5^\circ \sin 20^\circ - \cos 35^\circ \sin 50^\circ}
$$

---

### Step 1: Simplify the Numerator

We observe that the numerator can be grouped as:

$$
(\cos 5^\circ \cos 20^\circ - \sin 5^\circ \sin 20^\circ) + (\cos 35^\circ \cos 50^\circ - \sin 35^\circ \sin 50^\circ)
$$

Using the identity:

$$
\cos A \cos B - \sin A \sin B = \cos(A + B)
$$

We get:

$$
\cos(5^\circ + 20^\circ) + \cos(35^\circ + 50^\circ) = \cos 25^\circ + \cos 85^\circ
$$

---

### Step 2: Simplify the Denominator

The denominator is:

$$
(\sin 5^\circ \cos 20^\circ + \cos 5^\circ \sin 20^\circ) - (\sin 35^\circ \cos 50^\circ + \cos 35^\circ \sin 50^\circ)
$$

Using the identity:

$$
\sin A \cos B + \cos A \sin B = \sin(A + B)
$$

We get:

$$
\sin(5^\circ + 20^\circ) - \sin(35^\circ + 50^\circ) = \sin 25^\circ - \sin 85^\circ
$$

---

### Step 3: Apply Sum-to-Product Identities

Now we simplify both numerator and denominator using sum-to-product identities:

- **Numerator:**
  $$
  \cos 25^\circ + \cos 85^\circ = 2 \cos\left(\frac{25^\circ + 85^\circ}{2}\right) \cos\left(\frac{25^\circ - 85^\circ}{2}\right) = 2 \cos 55^\circ \cos(-30^\circ)
  $$
  Since $ \cos(-x) = \cos x $, this becomes:
  $$
  2 \cos 55^\circ \cos 30^\circ
  $$

- **Denominator:**
  $$
  \sin 25^\circ - \sin 85^\circ = 2 \cos\left(\frac{25^\circ + 85^\circ}{2}\right) \sin\left(\frac{25^\circ - 85^\circ}{2}\right) = 2 \cos 55^\circ \sin(-30^\circ)
  $$
  Since $ \sin(-x) = -\sin x $, this becomes:
  $$
  -2 \cos 55^\circ \sin 30^\circ
  $$

---

### Step 4: Simplify the Expression

Now, the expression becomes:

$$
\tan \theta = \frac{2 \cos 55^\circ \cos 30^\circ}{-2 \cos 55^\circ \sin 30^\circ}
$$

Cancel $ 2 \cos 55^\circ $ from numerator and denominator:

$$
\tan \theta = \frac{\cos 30^\circ}{-\sin 30^\circ} = -\cot 30^\circ
$$

Recall that $ \cot 30^\circ = \frac{1}{\tan 30^\circ} = \sqrt{3} $, so:

$$
\tan \theta = -\sqrt{3}
$$

---

### Step 5: Find the Least Positive Angle

We are to find the **least positive angle** $ \theta $ such that $ \tan \theta = -\sqrt{3} $.

The tangent function is negative in the **second and fourth quadrants**. The reference angle for $ \tan^{-1}(\sqrt{3}) = 60^\circ $, so:

- In the **second quadrant**: $ \theta = 180^\circ - 60^\circ = 120^\circ $
- In the **fourth quadrant**: $ \theta = 360^\circ - 60^\circ = 300^\circ $

Since we are looking for the **least positive** angle, the answer is:

$$
\boxed{120}
$$

Solution:
From the angle addition formula, the numerator is
\begin{align*}
&(\cos 5^\circ \cos 20^\circ - \sin 5^\circ \sin 20^\circ) + (\cos 35^\circ \cos 50^\circ -  \sin 35^\circ \sin 50^\circ) \\
&= \cos (5^\circ + 20^\circ) + \cos (35^\circ + 50^\circ) \\
&= \cos 25^\circ + \cos 85^\circ.
\end{align*}From the sum-to-product formula, $\cos 25^\circ + \cos 85^\circ = 2 \cos 55^\circ \cos 30^\circ.$

Similarly, the denominator is
\begin{align*}
&\sin 5^\circ \cos 20^\circ - \sin 35^\circ \cos 50^\circ + \cos 5^\circ \sin 20^\circ - \cos 35^\circ \sin 50^\circ) \\
&= (\sin 5^\circ \cos 20^\circ +  \cos 5^\circ \sin 20^\circ) - (\sin 35^\circ \cos 50^\circ + \cos 35^\circ \sin 50^\circ) \\
&= \sin (5^\circ + 20^\circ) - \sin (35^\circ + 50^\circ) \\
&= \sin 25^\circ - \sin 85^\circ \\
&= -2 \sin 30^\circ \cos 55^\circ,
\end{align*}so the expression is equal to
\[\frac{2 \cos 55^\circ \cos 30^\circ}{-2 \sin 30^\circ \cos 55^\circ} = -\frac{\cos 30^\circ}{\sin 30^\circ} = -\sqrt{3} = \tan 120^\circ.\]Hence, the smallest such $\theta$ is $\boxed{120^\circ}.$


Explanation:
Okay, so I need to find the number of ordered pairs (a, b) of integers such that the modulus of a + bi is less than or equal to 5. Hmm, modulus of a complex number a + bi is sqrt(a² + b²), right? So the condition |a + bi| ≤ 5 translates to sqrt(a² + b²) ≤ 5. If I square both sides to eliminate the square root, that becomes a² + b² ≤ 25. So the problem reduces to finding all integer pairs (a, b) where the sum of their squares is less than or equal to 25.

Alright, so I need to count all integer solutions (a, b) to a² + b² ≤ 25. Let me think about how to approach this. Maybe I can fix the value of a and then find the possible values of b for each a? That seems systematic. Let me try that.

First, note that a and b are integers, so they can be positive, negative, or zero. However, since a² and b² are non-negative, the values of a and b can range from -5 to 5, because if |a| or |b| is greater than 5, then a² or b² would be greater than 25, making a² + b² exceed 25. Wait, actually, if |a| is 5, then a² is 25, so b² has to be 0. Similarly, if |a| is 4, then a² is 16, so b² can be up to 9, meaning |b| can be up to 3. So maybe I can iterate over possible values of a from -5 to 5, and for each a, find the number of integers b such that b² ≤ 25 - a². Then sum all those counts.

Yes, that seems like a plan. Let me formalize this. For each integer a with |a| ≤ 5, compute the maximum integer b such that b² ≤ 25 - a². Then, for each such a, the number of possible b's is the number of integers between -floor(sqrt(25 - a²)) and floor(sqrt(25 - a²)), inclusive. Since for each a, the possible b's are symmetric around zero, except when b=0. So maybe for each a, the number of b's is 2*floor(sqrt(25 - a²)) + 1? Wait, let me check.

Suppose for a given a, the maximum value of |b| is k, so that k² ≤ 25 - a² < (k+1)². Then the number of integers b is 2k + 1, because b can be -k, -k+1, ..., 0, ..., k-1, k. So yes, that formula would work. Therefore, for each a, compute k = floor(sqrt(25 - a²)), then the number of b's is 2k + 1. Then sum over all a from -5 to 5.

But wait, let me check with an example. Let's take a = 0. Then 25 - a² = 25, so sqrt(25) = 5, so k = 5. Then number of b's is 2*5 + 1 = 11. Which makes sense: b can be from -5 to 5 inclusive, which is 11 numbers. Correct.

Another example: a = 1. Then 25 - 1 = 24. sqrt(24) is approx 4.898, so floor is 4. Then number of b's is 2*4 + 1 = 9. Let me check: b² ≤ 24. So |b| ≤ 4, since 4² = 16 ≤ 24 and 5² = 25 > 24. So b can be -4, -3, -2, -1, 0, 1, 2, 3, 4. That's 9 numbers. Correct. So the formula works.

Therefore, the strategy is:

1. For each integer a from -5 to 5 inclusive:

   a. Compute 25 - a².

   b. Compute k = floor(sqrt(25 - a²)).

   c. The number of b's is 2k + 1.

2. Sum all these numbers over a from -5 to 5.

But since the problem is symmetric in a and b, maybe there's a smarter way? Wait, but since we are considering all a from -5 to 5, and for each a, calculating the number of b's, maybe it's straightforward. However, since a ranges from -5 to 5, that's 11 values of a. Let me note that for each a, the number of b's depends only on |a|, because a² is same for a and -a. Therefore, maybe we can compute for a from 0 to 5, calculate the number of b's for each |a|, multiply by 2 (for positive and negative a) except when a=0. Then sum accordingly. That might save some computation.

Yes, that's a good idea. Let me try that approach.

So first, for a = 0: compute k = floor(sqrt(25 - 0)) = 5, number of b's is 11. Then for a = ±1, ±2, ..., ±5, each will have the same number of b's. So let me compute for a from 0 to 5, then multiply by 2 (except for a=0).

So let's proceed:

For a = 0:

Number of b's: 11.

For a = 1:

Compute 25 - 1² = 24. sqrt(24) ≈ 4.898, floor is 4. Number of b's: 2*4 + 1 = 9. Since a=1 and a=-1 both have same count, total for a=±1: 2*9 = 18.

For a = 2:

25 - 4 = 21. sqrt(21) ≈ 4.583, floor is 4. Number of b's: 9. So total for a=±2: 2*9 = 18.

For a = 3:

25 - 9 = 16. sqrt(16) = 4. So floor is 4. Number of b's: 9. Total for a=±3: 2*9 = 18.

Wait, hold on. Wait, sqrt(16) is exactly 4, so k=4. So same as before. So same count. But let me check. For a=3, a²=9, so b² ≤ 16. So |b| ≤ 4. So same as before. So yes, 9 values. So same as a=1,2.

For a=4:

25 - 16 = 9. sqrt(9) = 3. So k=3. Number of b's: 2*3 + 1 = 7. So for a=±4: 2*7 = 14.

For a=5:

25 - 25 = 0. sqrt(0) = 0. So k=0. Number of b's: 2*0 + 1 = 1. So for a=±5: 2*1 = 2.

So now, adding up all these:

a=0: 11

a=±1: 18

a=±2: 18

a=±3: 18

a=±4: 14

a=±5: 2

Total number of ordered pairs is 11 + 18 + 18 + 18 + 14 + 2.

Let me compute that step by step:

Start with 11 (for a=0).

Add 18 (a=±1): 11 + 18 = 29.

Add 18 (a=±2): 29 + 18 = 47.

Add 18 (a=±3): 47 + 18 = 65.

Add 14 (a=±4): 65 + 14 = 79.

Add 2 (a=±5): 79 + 2 = 81.

Wait, so total of 81 ordered pairs? Hmm. Let me check if that's correct. But let me verify with another approach to make sure.

Alternatively, maybe I can think of all the points (a, b) with a² + b² ≤ 25. This is like counting all integer lattice points inside or on a circle of radius 5 centered at the origin.

Alternatively, maybe I can compute the number of points for each radius squared from 0 to 25, and sum them up. But that might be more complicated. Alternatively, maybe I can check my previous calculation again.

Wait, let me check the counts again for each a:

For a=0: b ranges from -5 to 5, which is 11 numbers. Correct.

For a=±1: For each a=1 or -1, b² ≤ 24. So |b| ≤ 4. So b from -4 to 4, which is 9 numbers. So 9 per a, times 2 is 18. Correct.

For a=±2: same as above, since 25 - 4 = 21, which is still 4²=16 ≤ 21, so |b| ≤ 4. So same 9 numbers. So 18. Correct.

For a=±3: 25 - 9 = 16, so |b| ≤ 4. Same as above. So 9 numbers. 18. Correct.

For a=±4: 25 - 16 = 9, so |b| ≤ 3. So b from -3 to 3, which is 7 numbers. So 7 per a, times 2 is 14. Correct.

For a=±5: 25 - 25 = 0, so |b| ≤ 0, so b=0. So 1 per a, times 2 is 2. Correct.

So adding up: 11 + 18 + 18 + 18 + 14 + 2 = 81. So 81 ordered pairs. Hmm. But let me check with another approach.

Alternatively, think of all possible a and b such that a² + b² ≤ 25. Let me think of how many points there are in each quadrant and then multiply by 4, adjusting for axes and origin.

But maybe that's more complicated. Alternatively, maybe I can check for small values.

Alternatively, think of the total number of points as follows:

For each radius squared from 0 to 25, count the number of integer solutions.

But that might take longer. Alternatively, maybe I can check if 81 is the correct answer.

Wait, but let me think. For example, the number of points with a² + b² = 0 is 1 (only (0,0)).

For a² + b² = 1: (±1, 0), (0, ±1). So 4 points.

For a² + b² = 2: (±1, ±1). So 4 points.

For a² + b² = 4: (±2, 0), (0, ±2), (±1, ±√3) but √3 is not integer. Wait, so only (±2, 0), (0, ±2). So 4 points? Wait, but also (±√(4 - b²), ...). Wait, maybe I need to think again.

Wait, for a² + b² = 4, possible integer solutions:

a and b integers such that their squares add to 4. So possible combinations:

(±2, 0), (0, ±2), (±1, ±√(4 - 1)) but √3 is not integer. So only the four points above. So 4 points.

Similarly for a² + b² = 5: (±1, ±2), (±2, ±1). So 8 points.

Wait, but maybe this approach is getting too detailed. However, if I can compute the total number of points by summing over all possible radii, but that might take time. However, if I can check whether 81 is correct.

Alternatively, maybe I can think of the total number of points as follows:

For each a from -5 to 5, compute the number of b's as before. Wait, but that's exactly what I did earlier. So if that gives 81, maybe that's correct. But let me check for a=0, which gives 11 points. Then for a=±1, ±2, ±3, each gives 9, 9, 9, then for a=±4, 7, and a=±5, 1. So adding up:

For a=0: 11

For a=±1: 2*9 = 18

For a=±2: 2*9 = 18

For a=±3: 2*9 = 18

For a=±4: 2*7 = 14

For a=±5: 2*1 = 2

Total: 11 + 18 + 18 + 18 + 14 + 2 = 81. So that seems consistent.

But let me check with another approach. Let me think of the number of points in each quadrant and multiply by 4, then add the axes points.

But maybe that's more complicated. Alternatively, think of the total number of points as follows:

For each possible value of a from -5 to 5, compute the number of b's. Let me do that again, but this time for each a:

a = -5: 25 - (-5)^2 = 0, so b=0. So 1 value.

a = -4: 25 - 16 = 9, so b from -3 to 3: 7 values.

a = -3: 25 - 9 = 16, so b from -4 to 4: 9 values.

a = -2: 25 - 4 = 21, so b from -4 to 4: 9 values.

a = -1: 25 - 1 = 24, so b from -4 to 4: 9 values.

a = 0: 25 - 0 = 25, so b from -5 to 5: 11 values.

a = 1: same as a=-1: 9 values.

a = 2: same as a=-2: 9 values.

a = 3: same as a=-3: 9 values.

a = 4: same as a=-4: 7 values.

a = 5: same as a=-5: 1 value.

So adding them up:

For a=-5: 1

a=-4: 7

a=-3: 9

a=-2: 9

a=-1: 9

a=0: 11

a=1: 9

a=2: 9

a=3: 9

a=4: 7

a=5: 1

So adding them:

Start with a=-5: 1

a=-4: 1 + 7 = 8

a=-3: 8 + 9 = 17

a=-2: 17 + 9 = 26

a=-1: 26 + 9 = 35

a=0: 35 + 11 = 46

a=1: 46 + 9 = 55

a=2: 55 + 9 = 64

a=3: 64 + 9 = 73

a=4: 73 + 7 = 80

a=5: 80 + 1 = 81

Same result. So that seems to confirm that the total is 81. Therefore, the answer is 81? Wait, but let me check with another way. Maybe I can think of the number of points in each circle.

Alternatively, think of the number of integer points inside or on the circle of radius 5. There is a formula or known values? Well, for radius 5, the number of integer points is known? Maybe not, but let me check for smaller radii.

For example, radius 0: 1 point.

Radius 1: 1 (origin) + 4 (on the axes) = 5? Wait, no. Wait, for radius 1, the equation is a² + b² ≤ 1. So points are (0,0), (±1, 0), (0, ±1). So total 5 points. But according to our previous method, for a from -1 to 1:

For a=-1: b² ≤ 1 - 1 = 0 => b=0. So 1.

a=0: b from -1 to 1: 3.

a=1: same as a=-1: 1.

Total: 1 + 3 + 1 = 5. Correct.

Similarly, for radius 2, the number of points would be?

Using our method:

a from -2 to 2:

a=-2: b² ≤ 4 - 4 = 0 => 1.

a=-1: b² ≤ 4 - 1 = 3 => floor(sqrt(3))=1, so 2*1 +1=3.

a=0: b² ≤ 4 => floor(2)=2, so 5.

a=1: same as a=-1: 3.

a=2: same as a=-2: 1.

Total: 1 + 3 + 5 + 3 + 1 = 13.

Alternatively, manually: points with a² + b² ≤ 4.

All points where |a| ≤ 2, |b| ≤ 2, but with a² + b² ≤ 4.

So:

(0,0), (±1,0), (0, ±1), (±2, 0), (0, ±2), (±1, ±1), (±1, ±√(4 - 1)) but √3 is not integer. Wait, but (±1, ±1) gives 1 + 1 = 2 ≤ 4. So (±1, ±1) are allowed. Also (±1, ±2)? Wait, 1 + 4 = 5 > 4. So no. So points:

(0,0): 1

(±1, 0), (0, ±1): 4

(±2, 0), (0, ±2): 4

(±1, ±1): 4

Total: 1 + 4 + 4 + 4 = 13. Correct. So our method works here.

So for radius 2, 13 points. So our method is correct. Therefore, trusting that, for radius 5, 81 points. But let me check another radius. For example, radius sqrt(5), but maybe not necessary.

Alternatively, maybe I can check the total number of points for radius 5. Wait, but maybe I can think of the total number of points as follows:

The number of integer solutions to a² + b² ≤ 25 is equal to the sum_{k=0}^{25} r(k), where r(k) is the number of integer solutions to a² + b² = k. But calculating that might be tedious, but let me try for some values.

For example, for k=0: 1

k=1: 4

k=2: 4

k=4: 4

k=5: 8

k=8: 4

k=9: 4

k=10: 8

k=13: 8

k=16: 4

k=18: 4

k=20: 8

k=25: 4

Wait, but this is getting complicated. However, if I can compute the total number of points by adding up all these, but I need to know all the k's from 0 to 25. However, this might take too long. Alternatively, maybe I can check if 81 is a reasonable number.

Alternatively, think of the area of the circle. The area is πr² = 25π ≈ 78.54. Since the number of integer points should be roughly equal to the area, but since we are counting points inside or on the circle, it's possible that the number is around 80. So 81 is close to that. So that seems plausible.

Alternatively, maybe I can check for radius 5, the number of points is known. Wait, I recall that Gauss's circle problem states that the number of integer points inside a circle of radius r is approximately πr² + E, where E is the error term. For r=5, π*25 ≈ 78.54, and our count is 81, which is 2.46 more. Which seems reasonable.

Alternatively, maybe I can check OEIS or something, but since I can't access external resources, I need to rely on my calculation.

Alternatively, maybe I can think of the total number of points as follows:

For each a from -5 to 5, compute the number of b's:

For a=-5: 1

a=-4: 7

a=-3: 9

a=-2: 9

a=-1: 9

a=0: 11

a=1: 9

a=2: 9

a=3: 9

a=4: 7

a=5: 1

Adding them up:

Let me group them:

For a=-5 and a=5: 1 + 1 = 2

For a=-4 and a=4: 7 + 7 = 14

For a=-3, -2, -1, 1, 2, 3: each has 9, so 6*9 = 54

For a=0: 11

Total: 2 + 14 + 54 + 11 = 81. Yes, same result. So that seems correct.

Therefore, I think the answer is 81. However, just to be thorough, let me check for a specific a value.

Take a=3. Then a²=9, so b² ≤ 16. So |b| ≤ 4. So b can be -4, -3, -2, -1, 0, 1, 2, 3, 4. That's 9 values. Correct. Similarly, for a=4, b² ≤ 9, so |b| ≤ 3, which is 7 values. Correct. For a=0, 11 values. Correct. So all counts seem correct.

Therefore, I think the answer is 81. Wait, but let me check once more for possible miscalculations in addition.

Original breakdown:

a=-5: 1

a=-4: 7

a=-3: 9

a=-2: 9

a=-1: 9

a=0: 11

a=1: 9

a=2: 9

a=3: 9

a=4: 7

a=5: 1

Adding them:

Start with a=-5: 1

Add a=-4: 1 + 7 = 8

Add a=-3: 8 + 9 = 17

Add a=-2: 17 + 9 = 26

Add a=-1: 26 + 9 = 35

Add a=0: 35 + 11 = 46

Add a=1: 46 + 9 = 55

Add a=2: 55 + 9 = 64

Add a=3: 64 + 9 = 73

Add a=4: 73 + 7 = 80

Add a=5: 80 + 1 = 81. Yes, same as before.

Therefore, I think the answer is 81. However, just to be 100% sure, let me think of another way. For example, think of all possible pairs (a, b) where a and b are integers between -5 and 5 inclusive, and a² + b² ≤ 25. So total number of such pairs.

But how many total pairs are there? Well, if we consider all a and b from -5 to 5, that's 11*11=121 pairs. But not all of them satisfy a² + b² ≤ 25. So the number we are looking for is less than 121. Our answer is 81, which is less than 121, so that seems reasonable.

Alternatively, maybe I can compute how many pairs are excluded. But that might not be helpful. Alternatively, think of the number of points where a² + b² > 25. But again, not helpful.

Alternatively, think of the number of points on the circle a² + b² = 25. For radius 5, the integer solutions are (±5, 0), (0, ±5), (±3, ±4), (±4, ±3). So that's 4 + 4 + 8 = 16 points. Wait, so the number of points on the circle is 12? Wait:

(±5, 0): 2 points

(0, ±5): 2 points

(±3, ±4): 4 points (since both 3 and 4 can be positive or negative)

(±4, ±3): another 4 points. So total 2 + 2 + 4 + 4 = 12 points. So 12 points on the circle. Then, if we know the total number of points inside the circle (including boundary) is 81, then the number of points strictly inside would be 81 - 12 = 69. But I don't know if that helps.

Alternatively, if I can compute the number of points inside the circle (excluding boundary) and then add the boundary points. But again, not helpful.

Alternatively, maybe I can check for a smaller radius. For example, radius 1: total points 5, boundary points (on the circle) are 4 (the four points on the axes), and the origin is inside. So 5 total. Which matches.

For radius 2: total points 13. The boundary points are those with a² + b² = 4. Which are (±2, 0), (0, ±2), (±1, ±√(4 - 1)) but only integer points. Wait, (±2, 0), (0, ±2), and (±1, ±√3) which are not integers. So only 4 points on the boundary. Then the number of points inside would be 13 - 4 = 9. Which would be the points with a² + b² < 4. Which are the origin, (±1, 0), (0, ±1). So 1 + 4 = 5. Wait, but 13 - 4 = 9? That doesn't match. Wait, maybe my previous thought is wrong.

Wait, for radius 2, the total number of points with a² + b² ≤ 4 is 13. The number of points on the boundary (a² + b² = 4) is 4: (±2, 0), (0, ±2). Then the number of points strictly inside is 13 - 4 = 9. But the points strictly inside would be those with a² + b² < 4. Which are:

(0,0), (±1, 0), (0, ±1), (±1, ±1). So that's 1 + 4 + 4 = 9. Yes, that's correct. So 9 points inside, 4 on the boundary, and 1 origin? Wait, no. Wait, origin is included in the inside. So total points inside or on boundary is 9 + 4 = 13. Correct. So that works.

So for radius 5, if I can compute the number of points on the boundary (a² + b² = 25) which is 12, then the number of points inside would be 81 - 12 = 69. But again, not helpful for verification.

Alternatively, maybe I can check the total number of points for radius 5 by considering the number of points in each quadrant and multiplying by 4, then adding the axes points. Let me try that.

First, consider the first quadrant where a ≥ 0 and b ≥ 0. Then multiply by 4, but adjust for points on the axes.

But this might be complicated. Let me try.

In the first quadrant, a and b are non-negative integers with a² + b² ≤ 25. Let me count these points, then multiply by 4, subtract overlaps, etc.

But maybe it's too time-consuming, but let me try.

First quadrant points (including axes):

For a from 0 to 5:

For each a, find the number of b's such that b² ≤ 25 - a², with b ≥ 0.

So:

a=0: b from 0 to 5: 6 values.

a=1: b from 0 to 4 (since 4²=16 ≤ 24, 5²=25 > 24): 5 values.

a=2: b from 0 to 4 (since 25 - 4 = 21, sqrt(21) ≈ 4.58, so floor 4): 5 values.

a=3: b from 0 to 4 (25 - 9 = 16, sqrt(16)=4): 5 values.

a=4: b from 0 to 3 (25 - 16 = 9, sqrt(9)=3): 4 values.

a=5: b from 0 to 0 (25 - 25 = 0): 1 value.

So total in first quadrant: 6 + 5 + 5 + 5 + 4 + 1 = 26.

But wait, let me check:

a=0: 0-5: 6

a=1: 0-4: 5

a=2: 0-4: 5

a=3: 0-4: 5

a=4: 0-3: 4

a=5: 0-0: 1

Total: 6 + 5 + 5 + 5 + 4 + 1 = 26. Correct.

Now, the total number of points in all quadrants would be:

First quadrant points (including axes): 26.

But when we multiply by 4, we have to consider that points on the axes are counted multiple times. So:

- Points strictly inside the first quadrant (a > 0, b > 0): Let me compute that. For a from 1 to 5, and b from 1 to ... So total points in first quadrant excluding axes: 26 - (number of points on axes in first quadrant).

Number of points on axes in first quadrant: points where a=0 or b=0.

For a=0: b from 0 to 5: 6 points.

For b=0: a from 0 to 5: 6 points.

But the origin (0,0) is counted twice, so total points on axes in first quadrant: 6 + 6 - 1 = 11.

Therefore, points strictly inside first quadrant (a > 0, b > 0): 26 - 11 = 15.

Therefore, total number of points in all quadrants:

- Points strictly inside: 15 * 4 = 60.

- Points on the axes (but not origin): For each axis, there are points. Let me think.

Wait, maybe it's better to use inclusion-exclusion.

Total points in all quadrants:

= 4*(points strictly inside first quadrant) + 4*(points on axes in first quadrant excluding origin) + 1*(origin).

Wait, no. Let me think again.

The total number of points can be calculated as:

- 4*(number of points in first quadrant with a > 0, b > 0) + 2*(number of points on the axes excluding origin) + 1*(origin).

Wait, let me think:

- The points strictly inside the first quadrant (a > 0, b > 0) are 15. There are 4 such quadrants, so 15*4 = 60.

- The points on the axes (excluding origin) are:

For the positive x-axis: a from 1 to 5, b=0: 5 points.

Similarly, positive y-axis: b from 1 to 5, a=0: 5 points.

Similarly, negative x-axis: a from -1 to -5, b=0: 5 points.

Negative y-axis: b from -1 to -5, a=0: 5 points.

But wait, in the first quadrant, we already considered points on the axes. Wait, maybe this approach is getting too tangled.

Alternatively, since we have the first quadrant count as 26, which includes all points with a ≥ 0, b ≥ 0. Then, the total number of points in all quadrants would be:

- For points not on the axes: 4*(number of points in first quadrant with a > 0, b > 0) = 4*15 = 60.

- For points on the axes (excluding origin): each axis has 2*(number of points on positive axis) - 1 (to exclude origin). Wait, maybe not. Let me think.

In the first quadrant, the points on the axes are:

- On the x-axis: a from 0 to 5, b=0: 6 points.

- On the y-axis: b from 0 to 5, a=0: 6 points.

But the origin is counted in both. So total points on the axes in first quadrant: 6 + 6 - 1 = 11.

Therefore, total points on the axes in all quadrants would be:

- For x-axis: points (a, 0) where a from -5 to 5, excluding origin: 2*5 = 10.

- For y-axis: similarly 10.

- Plus the origin: 1.

Wait, but the total points on the axes (including origin) would be 2*11 - 1 = 21? Wait, no. Wait, in first quadrant, points on axes are 11. So in all quadrants, points on x-axis: 2* (points on positive x-axis) + origin? Wait, no. The points on the x-axis are (a, 0) where a ranges from -5 to 5. That's 11 points. Similarly for y-axis: 11 points. But the origin is counted in both. So total points on axes: 11 + 11 - 1 = 21.

Therefore, total number of points:

- Points strictly inside: 60

- Points on axes: 21

- Origin is included in points on axes.

Wait, but the origin is already counted in the axes. So total points would be 60 + 21 = 81. Which matches our previous result! So that's a good confirmation. Therefore, the total number of points is indeed 81. Therefore, the answer is 81.

Therefore, I can be confident that the number of ordered pairs (a, b) is 81.

**Final Answer**
\boxed{81}

To solve the problem of finding the number of ordered pairs $(a, b)$ of integers such that $|a + bi| \leq 5$, we begin by interpreting the condition in terms of the modulus of a complex number.

---

### Step 1: Translate the Condition

The modulus of a complex number $a + bi$ is given by:

$$
|a + bi| = \sqrt{a^2 + b^2}
$$

So the inequality $|a + bi| \leq 5$ becomes:

$$
\sqrt{a^2 + b^2} \leq 5
$$

Squaring both sides (since both sides are non-negative), we get:

$$
a^2 + b^2 \leq 25
$$

We are now looking for all integer pairs $(a, b)$ such that the sum of their squares is less than or equal to 25.

---

### Step 2: Strategy for Counting

We can count the number of integer solutions to the inequality $a^2 + b^2 \leq 25$ by iterating over all possible integer values of $a$ and, for each $a$, determining the number of integer values of $b$ that satisfy the inequality.

Let’s denote:

$$
k = \left\lfloor \sqrt{25 - a^2} \right\rfloor
$$

Then, for each fixed $a$, the number of integer values of $b$ such that $b^2 \leq 25 - a^2$ is:

$$
2k + 1
$$

This is because $b$ can range from $-k$ to $k$, inclusive, which gives $2k + 1$ values.

---

### Step 3: Compute the Total

We now compute the total number of such pairs by summing over all integer values of $a$ from $-5$ to $5$, since $a^2 \leq 25$ implies $|a| \leq 5$.

We can exploit symmetry: for each $a$, the number of valid $b$ values depends only on $|a|$, so we can compute for $a = 0$ to $5$, and multiply the result by 2 (except for $a = 0$).

Let’s compute:

- $a = 0$: $25 - 0 = 25$, so $k = \lfloor \sqrt{25} \rfloor = 5$, number of $b$ values = $2 \cdot 5 + 1 = 11$
- $a = \pm1$: $25 - 1 = 24$, $k = \lfloor \sqrt{24} \rfloor = 4$, number of $b$ values = $2 \cdot 4 + 1 = 9$
- $a = \pm2$: $25 - 4 = 21$, $k = \lfloor \sqrt{21} \rfloor = 4$, number of $b$ values = 9
- $a = \pm3$: $25 - 9 = 16$, $k = \lfloor \sqrt{16} \rfloor = 4$, number of $b$ values = 9
- $a = \pm4$: $25 - 16 = 9$, $k = \lfloor \sqrt{9} \rfloor = 3$, number of $b$ values = $2 \cdot 3 + 1 = 7$
- $a = \pm5$: $25 - 25 = 0$, $k = \lfloor \sqrt{0} \rfloor = 0$, number of $b$ values = $2 \cdot 0 + 1 = 1$

Now, summing up:

- $a = 0$: 11
- $a = \pm1$: $2 \cdot 9 = 18$
- $a = \pm2$: $2 \cdot 9 = 18$
- $a = \pm3$: $2 \cdot 9 = 18$
- $a = \pm4$: $2 \cdot 7 = 14$
- $a = \pm5$: $2 \cdot 1 = 2$

Total number of ordered pairs:

$$
11 + 18 + 18 + 18 + 14 + 2 = \boxed{81}
$$

---

### Final Answer

$$
\boxed{81}
$$

[Assistant]
Explanation:
Okay, so I need to find the number of ordered pairs (a, b) of integers such that the modulus of a + bi is less than or equal to 5. Hmm, modulus of a complex number a + bi is sqrt(a² + b²), right? So the condition |a + bi| ≤ 5 translates to sqrt(a² + b²) ≤ 5. If I square both sides to eliminate the square root, that becomes a² + b² ≤ 25. So the problem reduces to finding all integer pairs (a, b) where the sum of their squares is less than or equal to 25.

Alright, so I need to count all integer solutions (a, b) to a² + b² ≤ 25. Let me think about how to approach this. Maybe I can fix the value of a and then find the possible values of b for each a? That seems systematic. Let me try that.

First, note that a and b are integers, so they can be positive, negative, or zero. However, since a² and b² are non-negative, the values of a and b can range from -5 to 5, because if |a| or |b| is greater than 5, then a² or b² would be greater than 25, making a² + b² exceed 25. Wait, actually, if |a| is 5, then a² is 25, so b² has to be 0. Similarly, if |a| is 4, then a² is 16, so b² can be up to 9, meaning |b| can be up to 3. So maybe I can iterate over possible values of a from -5 to 5, and for each a, find the number of integers b such that b² ≤ 25 - a². Then sum all those counts.

Yes, that seems like a plan. Let me formalize this. For each integer a with |a| ≤ 5, compute the maximum integer b such that b² ≤ 25 - a². Then, for each such a, the number of possible b's is the number of integers between -floor(sqrt(25 - a²)) and floor(sqrt(25 - a²)), inclusive. Since for each a, the possible b's are symmetric around zero, except when b=0. So maybe for each a, the number of b's is 2*floor(sqrt(25 - a²)) + 1? Wait, let me check.

Suppose for a given a, the maximum value of |b| is k, so that k² ≤ 25 - a² < (k+1)². Then the number of integers b is 2k + 1, because b can be -k, -k+1, ..., 0, ..., k-1, k. So yes, that formula would work. Therefore, for each a, compute k = floor(sqrt(25 - a²)), then the number of b's is 2k + 1. Then sum over all a from -5 to 5.

But wait, let me check with an example. Let's take a = 0. Then 25 - a² = 25, so sqrt(25) = 5, so k = 5. Then number of b's is 2*5 + 1 = 11. Which makes sense: b can be from -5 to 5 inclusive, which is 11 numbers. Correct.

Another example: a = 1. Then 25 - 1 = 24. sqrt(24) is approx 4.898, so floor is 4. Then number of b's is 2*4 + 1 = 9. Let me check: b² ≤ 24. So |b| ≤ 4, since 4² = 16 ≤ 24 and 5² = 25 > 24. So b can be -4, -3, -2, -1, 0, 1, 2, 3, 4. That's 9 numbers. Correct. So the formula works.

Therefore, the strategy is:

1. For each integer a from -5 to 5 inclusive:

   a. Compute 25 - a².

   b. Compute k = floor(sqrt(25 - a²)).

   c. The number of b's is 2k + 1.

2. Sum all these numbers over a from -5 to 5.

But since the problem is symmetric in a and b, maybe there's a smarter way? Wait, but since we are considering all a from -5 to 5, and for each a, calculating the number of b's, maybe it's straightforward. However, since a ranges from -5 to 5, that's 11 values of a. Let me note that for each a, the number of b's depends only on |a|, because a² is same for a and -a. Therefore, maybe we can compute for a from 0 to 5, calculate the number of b's for each |a|, multiply by 2 (for positive and negative a) except when a=0. Then sum accordingly. That might save some computation.

Yes, that's a good idea. Let me try that approach.

So first, for a = 0: compute k = floor(sqrt(25 - 0)) = 5, number of b's is 11. Then for a = ±1, ±2, ..., ±5, each will have the same number of b's. So let me compute for a from 0 to 5, then multiply by 2 (except for a=0).

So let's proceed:

For a = 0:

Number of b's: 11.

For a = 1:

Compute 25 - 1² = 24. sqrt(24) ≈ 4.898, floor is 4. Number of b's: 2*4 + 1 = 9. Since a=1 and a=-1 both have same count, total for a=±1: 2*9 = 18.

For a = 2:

25 - 4 = 21. sqrt(21) ≈ 4.583, floor is 4. Number of b's: 9. So total for a=±2: 2*9 = 18.

For a = 3:

25 - 9 = 16. sqrt(16) = 4. So floor is 4. Number of b's: 9. Total for a=±3: 2*9 = 18.

Wait, hold on. Wait, sqrt(16) is exactly 4, so k=4. So same as before. So same count. But let me check. For a=3, a²=9, so b² ≤ 16. So |b| ≤ 4. So same as before. So yes, 9 values. So same as a=1,2.

For a=4:

25 - 16 = 9. sqrt(9) = 3. So k=3. Number of b's: 2*3 + 1 = 7. So for a=±4: 2*7 = 14.

For a=5:

25 - 25 = 0. sqrt(0) = 0. So k=0. Number of b's: 2*0 + 1 = 1. So for a=±5: 2*1 = 2.

So now, adding up all these:

a=0: 11

a=±1: 18

a=±2: 18

a=±3: 18

a=±4: 14

a=±5: 2

Total number of ordered pairs is 11 + 18 + 18 + 18 + 14 + 2.

Let me compute that step by step:

Start with 11 (for a=0).

Add 18 (a=±1): 11 + 18 = 29.

Add 18 (a=±2): 29 + 18 = 47.

Add 18 (a=±3): 47 + 18 = 65.

Add 14 (a=±4): 65 + 14 = 79.

Add 2 (a=±5): 79 + 2 = 81.

Wait, so total of 81 ordered pairs? Hmm. Let me check if that's correct. But let me verify with another approach to make sure.

Alternatively, maybe I can think of all the points (a, b) with a² + b² ≤ 25. This is like counting all integer lattice points inside or on a circle of radius 5 centered at the origin.

Alternatively, maybe I can compute the number of points for each radius squared from 0 to 25, and sum them up. But that might be more complicated. Alternatively, maybe I can check my previous calculation again.

Wait, let me check the counts again for each a:

For a=0: b ranges from -5 to 5, which is 11 numbers. Correct.

For a=±1: For each a=1 or -1, b² ≤ 24. So |b| ≤ 4. So b from -4 to 4, which is 9 numbers. So 9 per a, times 2 is 18. Correct.

For a=±2: same as above, since 25 - 4 = 21, which is still 4²=16 ≤ 21, so |b| ≤ 4. So same 9 numbers. So 18. Correct.

For a=±3: 25 - 9 = 16, so |b| ≤ 4. Same as above. So 9 numbers. 18. Correct.

For a=±4: 25 - 16 = 9, so |b| ≤ 3. So b from -3 to 3, which is 7 numbers. So 7 per a, times 2 is 14. Correct.

For a=±5: 25 - 25 = 0, so |b| ≤ 0, so b=0. So 1 per a, times 2 is 2. Correct.

So adding up: 11 + 18 + 18 + 18 + 14 + 2 = 81. So 81 ordered pairs. Hmm. But let me check with another approach.

Alternatively, think of all possible a and b such that a² + b² ≤ 25. Let me think of how many points there are in each quadrant and then multiply by 4, adjusting for axes and origin.

But maybe that's more complicated. Alternatively, maybe I can check for small values.

Alternatively, think of the total number of points as follows:

For each radius squared from 0 to 25, count the number of integer solutions.

But that might take longer. Alternatively, maybe I can check if 81 is the correct answer.

Wait, but let me think. For example, the number of points with a² + b² = 0 is 1 (only (0,0)).

For a² + b² = 1: (±1, 0), (0, ±1). So 4 points.

For a² + b² = 2: (±1, ±1). So 4 points.

For a² + b² = 4: (±2, 0), (0, ±2), (±1, ±√3) but √3 is not integer. Wait, so only (±2, 0), (0, ±2). So 4 points? Wait, but also (±√(4 - b²), ...). Wait, maybe I need to think again.

Wait, for a² + b² = 4, possible integer solutions:

a and b integers such that their squares add to 4. So possible combinations:

(±2, 0), (0, ±2), (±1, ±√(4 - 1)) but √3 is not integer. So only the four points above. So 4 points.

Similarly for a² + b² = 5: (±1, ±2), (±2, ±1). So 8 points.

Wait, but maybe this approach is getting too detailed. However, if I can compute the total number of points by summing over all possible radii, but that might take time. However, if I can check whether 81 is correct.

Alternatively, maybe I can think of the total number of points as follows:

For each a from -5 to 5, compute the number of b's as before. Wait, but that's exactly what I did earlier. So if that gives 81, maybe that's correct. But let me check for a=0, which gives 11 points. Then for a=±1, ±2, ±3, each gives 9, 9, 9, then for a=±4, 7, and a=±5, 1. So adding up:

For a=0: 11

For a=±1: 2*9 = 18

For a=±2: 2*9 = 18

For a=±3: 2*9 = 18

For a=±4: 2*7 = 14

For a=±5: 2*1 = 2

Total: 11 + 18 + 18 + 18 + 14 + 2 = 81. So that seems consistent.

But let me check with another approach. Let me think of the number of points in each quadrant and multiply by 4, then add the axes points.

But maybe that's more complicated. Alternatively, think of the total number of points as follows:

For each possible value of a from -5 to 5, compute the number of b's. Let me do that again, but this time for each a:

a = -5: 25 - (-5)^2 = 0, so b=0. So 1 value.

a = -4: 25 - 16 = 9, so b from -3 to 3: 7 values.

a = -3: 25 - 9 = 16, so b from -4 to 4: 9 values.

a = -2: 25 - 4 = 21, so b from -4 to 4: 9 values.

a = -1: 25 - 1 = 24, so b from -4 to 4: 9 values.

a = 0: 25 - 0 = 25, so b from -5 to 5: 11 values.

a = 1: same as a=-1: 9 values.

a = 2: same as a=-2: 9 values.

a = 3: same as a=-3: 9 values.

a = 4: same as a=-4: 7 values.

a = 5: same as a=-5: 1 value.

So adding them up:

For a=-5: 1

a=-4: 7

a=-3: 9

a=-2: 9

a=-1: 9

a=0: 11

a=1: 9

a=2: 9

a=3: 9

a=4: 7

a=5: 1

So adding them:

Start with a=-5: 1

a=-4: 1 + 7 = 8

a=-3: 8 + 9 = 17

a=-2: 17 + 9 = 26

a=-1: 26 + 9 = 35

a=0: 35 + 11 = 46

a=1: 46 + 9 = 55

a=2: 55 + 9 = 64

a=3: 64 + 9 = 73

a=4: 73 + 7 = 80

a=5: 80 + 1 = 81

Same result. So that seems to confirm that the total is 81. Therefore, the answer is 81? Wait, but let me check with another way. Maybe I can think of the number of points in each circle.

Alternatively, think of the number of integer points inside or on the circle of radius 5. There is a formula or known values? Well, for radius 5, the number of integer points is known? Maybe not, but let me check for smaller radii.

For example, radius 0: 1 point.

Radius 1: 1 (origin) + 4 (on the axes) = 5? Wait, no. Wait, for radius 1, the equation is a² + b² ≤ 1. So points are (0,0), (±1, 0), (0, ±1). So total 5 points. But according to our previous method, for a from -1 to 1:

For a=-1: b² ≤ 1 - 1 = 0 => b=0. So 1.

a=0: b from -1 to 1: 3.

a=1: same as a=-1: 1.

Total: 1 + 3 + 1 = 5. Correct.

Similarly, for radius 2, the number of points would be?

Using our method:

a from -2 to 2:

a=-2: b² ≤ 4 - 4 = 0 => 1.

a=-1: b² ≤ 4 - 1 = 3 => floor(sqrt(3))=1, so 2*1 +1=3.

a=0: b² ≤ 4 => floor(2)=2, so 5.

a=1: same as a=-1: 3.

a=2: same as a=-2: 1.

Total: 1 + 3 + 5 + 3 + 1 = 13.

Alternatively, manually: points with a² + b² ≤ 4.

All points where |a| ≤ 2, |b| ≤ 2, but with a² + b² ≤ 4.

So:

(0,0), (±1,0), (0, ±1), (±2, 0), (0, ±2), (±1, ±1), (±1, ±√(4 - 1)) but √3 is not integer. Wait, but (±1, ±1) gives 1 + 1 = 2 ≤ 4. So (±1, ±1) are allowed. Also (±1, ±2)? Wait, 1 + 4 = 5 > 4. So no. So points:

(0,0): 1

(±1, 0), (0, ±1): 4

(±2, 0), (0, ±2): 4

(±1, ±1): 4

Total: 1 + 4 + 4 + 4 = 13. Correct. So our method works here.

So for radius 2, 13 points. So our method is correct. Therefore, trusting that, for radius 5, 81 points. But let me check another radius. For example, radius sqrt(5), but maybe not necessary.

Alternatively, maybe I can check the total number of points for radius 5. Wait, but maybe I can think of the total number of points as follows:

The number of integer solutions to a² + b² ≤ 25 is equal to the sum_{k=0}^{25} r(k), where r(k) is the number of integer solutions to a² + b² = k. But calculating that might be tedious, but let me try for some values.

For example, for k=0: 1

k=1: 4

k=2: 4

k=4: 4

k=5: 8

k=8: 4

k=9: 4

k=10: 8

k=13: 8

k=16: 4

k=18: 4

k=20: 8

k=25: 4

Wait, but this is getting complicated. However, if I can compute the total number of points by adding up all these, but I need to know all the k's from 0 to 25. However, this might take too long. Alternatively, maybe I can check if 81 is a reasonable number.

Alternatively, think of the area of the circle. The area is πr² = 25π ≈ 78.54. Since the number of integer points should be roughly equal to the area, but since we are counting points inside or on the circle, it's possible that the number is around 80. So 81 is close to that. So that seems plausible.

Alternatively, maybe I can check for radius 5, the number of points is known. Wait, I recall that Gauss's circle problem states that the number of integer points inside a circle of radius r is approximately πr² + E, where E is the error term. For r=5, π*25 ≈ 78.54, and our count is 81, which is 2.46 more. Which seems reasonable.

Alternatively, maybe I can check OEIS or something, but since I can't access external resources, I need to rely on my calculation.

Alternatively, maybe I can think of the total number of points as follows:

For each a from -5 to 5, compute the number of b's:

For a=-5: 1

a=-4: 7

a=-3: 9

a=-2: 9

a=-1: 9

a=0: 11

a=1: 9

a=2: 9

a=3: 9

a=4: 7

a=5: 1

Adding them up:

Let me group them:

For a=-5 and a=5: 1 + 1 = 2

For a=-4 and a=4: 7 + 7 = 14

For a=-3, -2, -1, 1, 2, 3: each has 9, so 6*9 = 54

For a=0: 11

Total: 2 + 14 + 54 + 11 = 81. Yes, same result. So that seems correct.

Therefore, I think the answer is 81. However, just to be thorough, let me check for a specific a value.

Take a=3. Then a²=9, so b² ≤ 16. So |b| ≤ 4. So b can be -4, -3, -2, -1, 0, 1, 2, 3, 4. That's 9 values. Correct. Similarly, for a=4, b² ≤ 9, so |b| ≤ 3, which is 7 values. Correct. For a=0, 11 values. Correct. So all counts seem correct.

Therefore, I think the answer is 81. Wait, but let me check once more for possible miscalculations in addition.

Original breakdown:

a=-5: 1

a=-4: 7

a=-3: 9

a=-2: 9

a=-1: 9

a=0: 11

a=1: 9

a=2: 9

a=3: 9

a=4: 7

a=5: 1

Adding them:

Start with a=-5: 1

Add a=-4: 1 + 7 = 8

Add a=-3: 8 + 9 = 17

Add a=-2: 17 + 9 = 26

Add a=-1: 26 + 9 = 35

Add a=0: 35 + 11 = 46

Add a=1: 46 + 9 = 55

Add a=2: 55 + 9 = 64

Add a=3: 64 + 9 = 73

Add a=4: 73 + 7 = 80

Add a=5: 80 + 1 = 81. Yes, same as before.

Therefore, I think the answer is 81. However, just to be 100% sure, let me think of another way. For example, think of all possible pairs (a, b) where a and b are integers between -5 and 5 inclusive, and a² + b² ≤ 25. So total number of such pairs.

But how many total pairs are there? Well, if we consider all a and b from -5 to 5, that's 11*11=121 pairs. But not all of them satisfy a² + b² ≤ 25. So the number we are looking for is less than 121. Our answer is 81, which is less than 121, so that seems reasonable.

Alternatively, maybe I can compute how many pairs are excluded. But that might not be helpful. Alternatively, think of the number of points where a² + b² > 25. But again, not helpful.

Alternatively, think of the number of points on the circle a² + b² = 25. For radius 5, the integer solutions are (±5, 0), (0, ±5), (±3, ±4), (±4, ±3). So that's 4 + 4 + 8 = 16 points. Wait, so the number of points on the circle is 12? Wait:

(±5, 0): 2 points

(0, ±5): 2 points

(±3, ±4): 4 points (since both 3 and 4 can be positive or negative)

(±4, ±3): another 4 points. So total 2 + 2 + 4 + 4 = 12 points. So 12 points on the circle. Then, if we know the total number of points inside the circle (including boundary) is 81, then the number of points strictly inside would be 81 - 12 = 69. But I don't know if that helps.

Alternatively, if I can compute the number of points inside the circle (excluding boundary) and then add the boundary points. But again, not helpful.

Alternatively, maybe I can check for a smaller radius. For example, radius 1: total points 5, boundary points (on the circle) are 4 (the four points on the axes), and the origin is inside. So 5 total. Which matches.

For radius 2: total points 13. The boundary points are those with a² + b² = 4. Which are (±2, 0), (0, ±2), (±1, ±√(4 - 1)) but only integer points. Wait, (±2, 0), (0, ±2), and (±1, ±√3) which are not integers. So only 4 points on the boundary. Then the number of points inside would be 13 - 4 = 9. Which would be the points with a² + b² < 4. Which are the origin, (±1, 0), (0, ±1). So 1 + 4 = 5. Wait, but 13 - 4 = 9? That doesn't match. Wait, maybe my previous thought is wrong.

Wait, for radius 2, the total number of points with a² + b² ≤ 4 is 13. The number of points on the boundary (a² + b² = 4) is 4: (±2, 0), (0, ±2). Then the number of points strictly inside is 13 - 4 = 9. But the points strictly inside would be those with a² + b² < 4. Which are:

(0,0), (±1, 0), (0, ±1), (±1, ±1). So that's 1 + 4 + 4 = 9. Yes, that's correct. So 9 points inside, 4 on the boundary, and 1 origin? Wait, no. Wait, origin is included in the inside. So total points inside or on boundary is 9 + 4 = 13. Correct. So that works.

So for radius 5, if I can compute the number of points on the boundary (a² + b² = 25) which is 12, then the number of points inside would be 81 - 12 = 69. But again, not helpful for verification.

Alternatively, maybe I can check the total number of points for radius 5 by considering the number of points in each quadrant and multiplying by 4, then adding the axes points. Let me try that.

First, consider the first quadrant where a ≥ 0 and b ≥ 0. Then multiply by 4, but adjust for points on the axes.

But this might be complicated. Let me try.

In the first quadrant, a and b are non-negative integers with a² + b² ≤ 25. Let me count these points, then multiply by 4, subtract overlaps, etc.

But maybe it's too time-consuming, but let me try.

First quadrant points (including axes):

For a from 0 to 5:

For each a, find the number of b's such that b² ≤ 25 - a², with b ≥ 0.

So:

a=0: b from 0 to 5: 6 values.

a=1: b from 0 to 4 (since 4²=16 ≤ 24, 5²=25 > 24): 5 values.

a=2: b from 0 to 4 (since 25 - 4 = 21, sqrt(21) ≈ 4.58, so floor 4): 5 values.

a=3: b from 0 to 4 (25 - 9 = 16, sqrt(16)=4): 5 values.

a=4: b from 0 to 3 (25 - 16 = 9, sqrt(9)=3): 4 values.

a=5: b from 0 to 0 (25 - 25 = 0): 1 value.

So total in first quadrant: 6 + 5 + 5 + 5 + 4 + 1 = 26.

But wait, let me check:

a=0: 0-5: 6

a=1: 0-4: 5

a=2: 0-4: 5

a=3: 0-4: 5

a=4: 0-3: 4

a=5: 0-0: 1

Total: 6 + 5 + 5 + 5 + 4 + 1 = 26. Correct.

Now, the total number of points in all quadrants would be:

First quadrant points (including axes): 26.

But when we multiply by 4, we have to consider that points on the axes are counted multiple times. So:

- Points strictly inside the first quadrant (a > 0, b > 0): Let me compute that. For a from 1 to 5, and b from 1 to ... So total points in first quadrant excluding axes: 26 - (number of points on axes in first quadrant).

Number of points on axes in first quadrant: points where a=0 or b=0.

For a=0: b from 0 to 5: 6 points.

For b=0: a from 0 to 5: 6 points.

But the origin (0,0) is counted twice, so total points on axes in first quadrant: 6 + 6 - 1 = 11.

Therefore, points strictly inside first quadrant (a > 0, b > 0): 26 - 11 = 15.

Therefore, total number of points in all quadrants:

- Points strictly inside: 15 * 4 = 60.

- Points on the axes (but not origin): For each axis, there are points. Let me think.

Wait, maybe it's better to use inclusion-exclusion.

Total points in all quadrants:

= 4*(points strictly inside first quadrant) + 4*(points on axes in first quadrant excluding origin) + 1*(origin).

Wait, no. Let me think again.

The total number of points can be calculated as:

- 4*(number of points in first quadrant with a > 0, b > 0) + 2*(number of points on the axes excluding origin) + 1*(origin).

Wait, let me think:

- The points strictly inside the first quadrant (a > 0, b > 0) are 15. There are 4 such quadrants, so 15*4 = 60.

- The points on the axes (excluding origin) are:

For the positive x-axis: a from 1 to 5, b=0: 5 points.

Similarly, positive y-axis: b from 1 to 5, a=0: 5 points.

Similarly, negative x-axis: a from -1 to -5, b=0: 5 points.

Negative y-axis: b from -1 to -5, a=0: 5 points.

But wait, in the first quadrant, we already considered points on the axes. Wait, maybe this approach is getting too tangled.

Alternatively, since we have the first quadrant count as 26, which includes all points with a ≥ 0, b ≥ 0. Then, the total number of points in all quadrants would be:

- For points not on the axes: 4*(number of points in first quadrant with a > 0, b > 0) = 4*15 = 60.

- For points on the axes (excluding origin): each axis has 2*(number of points on positive axis) - 1 (to exclude origin). Wait, maybe not. Let me think.

In the first quadrant, the points on the axes are:

- On the x-axis: a from 0 to 5, b=0: 6 points.

- On the y-axis: b from 0 to 5, a=0: 6 points.

But the origin is counted in both. So total points on the axes in first quadrant: 6 + 6 - 1 = 11.

Therefore, total points on the axes in all quadrants would be:

- For x-axis: points (a, 0) where a from -5 to 5, excluding origin: 2*5 = 10.

- For y-axis: similarly 10.

- Plus the origin: 1.

Wait, but the total points on the axes (including origin) would be 2*11 - 1 = 21? Wait, no. Wait, in first quadrant, points on axes are 11. So in all quadrants, points on x-axis: 2* (points on positive x-axis) + origin? Wait, no. The points on the x-axis are (a, 0) where a ranges from -5 to 5. That's 11 points. Similarly for y-axis: 11 points. But the origin is counted in both. So total points on axes: 11 + 11 - 1 = 21.

Therefore, total number of points:

- Points strictly inside: 60

- Points on axes: 21

- Origin is included in points on axes.

Wait, but the origin is already counted in the axes. So total points would be 60 + 21 = 81. Which matches our previous result! So that's a good confirmation. Therefore, the total number of points is indeed 81. Therefore, the answer is 81.

Therefore, I can be confident that the number of ordered pairs (a, b) is 81.

**Final Answer**
\boxed{81}

To solve the problem of finding the number of ordered pairs $(a, b)$ of integers such that $|a + bi| \leq 5$, we begin by interpreting the condition in terms of the modulus of a complex number.

---

### Step 1: Translate the Condition

The modulus of a complex number $a + bi$ is given by:

$$
|a + bi| = \sqrt{a^2 + b^2}
$$

So the inequality $|a + bi| \leq 5$ becomes:

$$
\sqrt{a^2 + b^2} \leq 5
$$

Squaring both sides (since both sides are non-negative), we get:

$$
a^2 + b^2 \leq 25
$$

We are now looking for all integer pairs $(a, b)$ such that the sum of their squares is less than or equal to 25.

---

### Step 2: Strategy for Counting

We can count the number of integer solutions to the inequality $a^2 + b^2 \leq 25$ by iterating over all possible integer values of $a$ and, for each $a$, determining the number of integer values of $b$ that satisfy the inequality.

Let’s denote:

$$
k = \left\lfloor \sqrt{25 - a^2} \right\rfloor
$$

Then, for each fixed $a$, the number of integer values of $b$ such that $b^2 \leq 25 - a^2$ is:

$$
2k + 1
$$

This is because $b$ can range from $-k$ to $k$, inclusive, which gives $2k + 1$ values.

---

### Step 3: Compute the Total

We now compute the total number of such pairs by summing over all integer values of $a$ from $-5$ to $5$, since $a^2 \leq 25$ implies $|a| \leq 5$.

We can exploit symmetry: for each $a$, the number of valid $b$ values depends only on $|a|$, so we can compute for $a = 0$ to $5$, and multiply the result by 2 (except for $a = 0$).

Let’s compute:

- $a = 0$: $25 - 0 = 25$, so $k = \lfloor \sqrt{25} \rfloor = 5$, number of $b$ values = $2 \cdot 5 + 1 = 11$
- $a = \pm1$: $25 - 1 = 24$, $k = \lfloor \sqrt{24} \rfloor = 4$, number of $b$ values = $2 \cdot 4 + 1 = 9$
- $a = \pm2$: $25 - 4 = 21$, $k = \lfloor \sqrt{21} \rfloor = 4$, number of $b$ values = 9
- $a = \pm3$: $25 - 9 = 16$, $k = \lfloor \sqrt{16} \rfloor = 4$, number of $b$ values = 9
- $a = \pm4$: $25 - 16 = 9$, $k = \lfloor \sqrt{9} \rfloor = 3$, number of $b$ values = $2 \cdot 3 + 1 = 7$
- $a = \pm5$: $25 - 25 = 0$, $k = \lfloor \sqrt{0} \rfloor = 0$, number of $b$ values = $2 \cdot 0 + 1 = 1$

Now, summing up:

- $a = 0$: 11
- $a = \pm1$: $2 \cdot 9 = 18$
- $a = \pm2$: $2 \cdot 9 = 18$
- $a = \pm3$: $2 \cdot 9 = 18$
- $a = \pm4$: $2 \cdot 7 = 14$
- $a = \pm5$: $2 \cdot 1 = 2$

Total number of ordered pairs:

$$
11 + 18 + 18 + 18 + 14 + 2 = \boxed{81}
$$

---

### Final Answer

$$
\boxed{81}
$$

Solution:
The problem asks us to count the number of complex numbers that lie in or on the circle of radius 5 centered at the origin, with integer real and imaginary parts.

[asy]
unitsize(0.5 cm);

int i, j;

draw((-5,0)--(5,0));
draw((0,-5)--(0,5));
draw(Circle((0,0),5));

for (i = -5; i <= 5; ++i) {
for (j = -5; j <= 5; ++j) {
  if (i^2 + j^2 > 25) {dot((i,j));}
  if (i^2 + j^2 <= 25) {dot((i,j),red);}
}}
[/asy]

We can count that there are 15 such complex numbers in the first quadrant (not including the axes).  Then there are 5 complex on the positive real axis, the negative real axis, the positive imaginary axis, and negative imaginary axis.  Finally, there is the origin itself, which gives us $4 \cdot 15 + 4 \cdot 5 + 1 = \boxed{81}$ complex numbers.