Explanation:
Okay, let me try to figure out this problem. So, the question is about a book that has a certain number of leaves, which is the number of pieces of paper. Each leaf has two pages, so the number of pages is twice the number of leaves. The problem says that the number of pages is 3 more than a multiple of 7. Also, the number of leaves is greater than 100. We need to find the smallest possible number of leaves that satisfies these conditions.

First, let me make sure I understand the terms correctly. A leaf is a single sheet of paper, right? So, if a book has n leaves, that means it has n sheets of paper. But each sheet has two pages: one on the front and one on the back. So, the total number of pages would be 2n. Yeah, that seems right. So, pages = 2 * leaves.

The problem states that the number of pages is 3 more than a multiple of 7. So, mathematically, that would mean pages ≡ 3 mod 7. Since pages = 2n, then 2n ≡ 3 mod 7. Therefore, we need to find the smallest n > 100 such that 2n ≡ 3 mod 7.

But wait, let me check that again. If pages = 2n, then 2n ≡ 3 mod 7. So, solving for n here. Let me write that as an equation: 2n ≡ 3 mod 7. To solve for n, we can multiply both sides by the modular inverse of 2 mod 7. The inverse of 2 mod 7 is a number x such that 2x ≡ 1 mod 7. Let me calculate that. Testing numbers:

2*1=2≡2

2*2=4≡4

2*3=6≡6

2*4=8≡1 mod7. Ah, so 4 is the inverse of 2 mod7.

Therefore, multiplying both sides by 4: n ≡ 4*3 mod7 => n ≡ 12 mod7 => 12 divided by 7 is 1 with remainder 5, so n ≡5 mod7.

Therefore, the number of leaves n must be congruent to 5 mod7, and n >100. We need the smallest such n.

So, the question becomes: find the smallest integer n >100 such that n ≡5 mod7.

So, let's think. The numbers congruent to 5 mod7 are numbers like 5, 12, 19, 26,... each time adding 7. So, starting from 5, add 7 repeatedly.

We need the smallest one greater than 100. Let me find the first n >100 where n ≡5 mod7.

Let me compute 100 divided by 7 to find where 100 falls in the sequence.

7*14=98, 7*14+2=100. So, 100 is 7*14 + 2. So, the next number congruent to 5 mod7 after 100 would be?

Let me think. Let me find the smallest number greater than 100 which is 5 mod7.

Let me compute 100 mod7. Since 7*14=98, 100-98=2. Therefore, 100 ≡2 mod7. So we need to find n such that n ≡5 mod7, and n>100.

So, starting from 100, we can add (5 - 2) mod7? Wait, maybe another way. Let me think.

Let me find how much I need to add to 100 to get to the next number that is 5 mod7.

Since 100 ≡2 mod7. The difference between 5 and 2 is 3, so if I add 3 to 100, then 100 +3=103. But 103 mod7 would be (2 +3)=5 mod7. So 103 ≡5 mod7. Therefore, 103 is the first number after 100 that is congruent to 5 mod7. But wait, is 103 greater than 100? Yes. So is 103 the minimal n?

Wait, but let me check. Let me check what's 7*14=98, then 98+5=103. So 103 is the first number in the sequence 5,12,...,103,... that is larger than 100. Therefore, n=103.

But hold on, is 103 the minimal? Let's check numbers between 100 and 103. For example, 101: 101-98=3, so 3 mod7. 102: 102-98=4 mod7. 103: 103-98=5 mod7. So yes, 103 is the first one. Therefore, the minimal number of leaves is 103.

But wait, let me check if that satisfies the original condition. The number of pages is 2*103=206. Is 206 3 more than a multiple of 7?

Let me compute 206 divided by 7. 7*29=203, 203+3=206. So yes, 206=7*29 +3. Therefore, 206 is indeed 3 more than a multiple of 7. Therefore, 103 leaves would make 206 pages, which satisfies the condition, and 103 is greater than 100.

But the question is, is 103 the minimal? Let me check if there is a number between 100 and 103 that is congruent to 5 mod7. Since 100 ≡2 mod7, then numbers after 100:

100 ≡2

101≡3

102≡4

103≡5

So, yes, 103 is the first one. Therefore, 103 is the minimal number of leaves.

But wait, the problem says "the number of leaves is greater than 100". So leaves must be greater than 100, so n>100. Therefore, 103 is indeed the answer. So why is this question considered challenging? Maybe I made a mistake?

Wait, let me check again. The number of pages is twice the number of leaves, so if leaves is n, pages is 2n. The problem says pages is 3 more than a multiple of 7. So 2n ≡3 mod7.

Earlier, I solved for n ≡ (3 * inverse of 2) mod7. Since inverse of 2 mod7 is 4, so 3*4=12≡5 mod7. Therefore, n≡5 mod7. Therefore, n=7k +5, for some integer k.

So n>100, so 7k +5>100 => 7k>95 =>k>95/7≈13.571. So k≥14.

Thus, the minimal k is 14, so n=7*14 +5=98 +5=103. Yep, same answer. So that seems correct.

Wait, but let me check if n=103 is the minimal. If k=14 gives n=103, then k=13 would be 7*13 +5=91 +5=96, which is less than 100. So yes, 103 is the first n>100. Therefore, 103 is the answer.

But let me check once more. Suppose n=103. Pages=206. 206 divided by 7: 7*29=203, 206-203=3. Yes, 206=7*29 +3. Therefore, 3 more than multiple of 7. So that works. And leaves=103>100. So that's the minimal.

But wait, hold on. Maybe there's another way that the pages could be 3 more than multiple of 7? For example, if pages=7m +3. Since pages=2n, then 2n=7m +3. Therefore, 2n ≡3 mod7. Which is what we had before. So solving for n, which gives n≡5 mod7. So same result.

Therefore, seems like 103 is the answer.

But hold on, I need to make sure that n must be integer. Since number of leaves has to be integer. Well, yes, n is integer. So, 103 is integer. So that's okay.

Alternatively, is there a case where maybe the number of leaves is not an integer? No, because leaves are pieces of paper, so they have to be whole numbers. So n must be integer.

Therefore, I think the answer is 103. However, let me check if there is a smaller n>100 that satisfies 2n ≡3 mod7. Suppose someone thought that maybe pages is 3 more than a multiple of 7, so pages=7k +3, and pages=2n. So, 2n=7k +3. Therefore, n=(7k +3)/2. Therefore, for n to be integer, 7k +3 must be even. Therefore, 7k must be odd, since 7k +3 is even. Therefore, 7k is odd. Since 7 is odd, k must be odd. Therefore, k is odd. Therefore, k=2m +1 for some integer m.

Therefore, substituting back, n=(7*(2m +1) +3)/2=(14m +7 +3)/2=(14m +10)/2=7m +5. Therefore, n=7m +5. Which is same as before. Therefore, n must be congruent to 5 mod7. Therefore, the minimal n>100 is 103. Therefore, same conclusion.

Alternatively, perhaps I can think of it as looking for numbers n>100 where n ≡5 mod7. The numbers would be 103, 110, 117,... So 103 is the first one. Therefore, answer is 103.

Wait, but let me check once again with another approach. Let me consider the equation 2n ≡3 mod7. So, solving for n:

Multiply both sides by inverse of 2 mod7 which is 4, so n ≡ 3*4 mod7 => n≡12 mod7 => n≡5 mod7. So same result.

Therefore, n=7k +5. Then, find minimal k such that 7k +5>100. So, 7k>95 =>k>95/7≈13.571. Therefore, k=14. Therefore, n=7*14 +5=98 +5=103. So same answer.

So, seems like 103 is the answer. But maybe the problem is in the definition of a leaf? Wait, the problem says "a book is said to have n leaves if it is composed of n pieces of paper." So, that is, one leaf is one piece of paper. So, each leaf is one sheet, which has two pages. Therefore, number of pages is 2n. So, that seems correct.

But let me check for example, if a book has 1 leaf, that is 1 sheet of paper, so pages is 2. So 2 pages. So that seems correct. So yes, pages=2n.

Therefore, with that, the problem is solved. So, answer is 103. Therefore, I think that the answer is 103. Therefore, the minimal number of leaves is 103.

But to double-check, let me see if there is a possible mistake here. Let me check the value of n=103. Then pages=206. 206 divided by 7 is 29 with remainder 3. 7*29=203, 206-203=3. So yes, 3 more than a multiple of 7. And 103>100. So that works. Therefore, 103 is correct.

But just to be thorough, let me check the next lower number that is 5 mod7. For example, n=103-7=96. But 96 is less than 100, so not allowed. Hence, 103 is the first one above 100. Therefore, answer is 103.

So I think that is correct.

**Final Answer**
The smallest possible number of leaves is \boxed{103}.

To solve this problem, we need to find the **smallest number of leaves** $ n $ such that:

1. The number of **pages** in the book is **3 more than a multiple of 7**, and
2. The number of **leaves** is **greater than 100**.

---

### Step 1: Understand the Relationship Between Leaves and Pages

We are told that:
- The number of **pages** in the book is **twice** the number of **leaves**, i.e.,
  $$
  \text{pages} = 2n
  $$

Also, the number of pages is said to be **3 more than a multiple of 7**, i.e.,
$$
2n \equiv 3 \pmod{7}
$$

---

### Step 2: Solve the Congruence

We are solving:
$$
2n \equiv 3 \pmod{7}
$$

To eliminate the coefficient 2 on the left, we find the **modular inverse** of 2 modulo 7. The inverse of 2 mod 7 is the number $ x $ such that:
$$
2x \equiv 1 \pmod{7}
$$

Testing small integers:
- $ 2 \cdot 4 = 8 \equiv 1 \pmod{7} $

So, inverse of 2 mod 7 is 4. Multiply both sides of the congruence by 4:
$$
4 \cdot (2n) \equiv 4 \cdot 3 \pmod{7}
\Rightarrow 8n \equiv 12 \pmod{7}
$$
$$
8n \equiv 12 \pmod{7} \Rightarrow n \equiv 5 \pmod{7}
$$

So, the number of **leaves** must satisfy:
$$
n \equiv 5 \pmod{7}
$$

---

### Step 3: Find the Smallest $ n > 100 $ That Satisfies This

We now find the **smallest integer** $ n > 100 $ such that:
$$
n \equiv 5 \pmod{7}
$$

We can write $ n = 7k + 5 $, and solve for $ k $ such that:
$$
7k + 5 > 100 \Rightarrow 7k > 95 \Rightarrow k > \frac{95}{7} \approx 13.57
$$

So the smallest integer $ k $ satisfying this is $ k = 14 $. Then:
$$
n = 7 \cdot 14 + 5 = 98 + 5 = 103
$$

---

### Step 4: Verify the Solution

Check the number of pages:
$$
\text{pages} = 2n = 2 \cdot 103 = 206
$$

Check if 206 is 3 more than a multiple of 7:
$$
206 \div 7 = 29 \text{ remainder } 3 \Rightarrow 206 = 7 \cdot 29 + 3
$$

So, 206 is indeed 3 more than a multiple of 7. Also, $ n = 103 > 100 $.

---

### Final Answer

$$
\boxed{103}
$$

[Assistant]
Explanation:
Okay, let me try to figure out this problem. So, the question is about a book that has a certain number of leaves, which is the number of pieces of paper. Each leaf has two pages, so the number of pages is twice the number of leaves. The problem says that the number of pages is 3 more than a multiple of 7. Also, the number of leaves is greater than 100. We need to find the smallest possible number of leaves that satisfies these conditions.

First, let me make sure I understand the terms correctly. A leaf is a single sheet of paper, right? So, if a book has n leaves, that means it has n sheets of paper. But each sheet has two pages: one on the front and one on the back. So, the total number of pages would be 2n. Yeah, that seems right. So, pages = 2 * leaves.

The problem states that the number of pages is 3 more than a multiple of 7. So, mathematically, that would mean pages ≡ 3 mod 7. Since pages = 2n, then 2n ≡ 3 mod 7. Therefore, we need to find the smallest n > 100 such that 2n ≡ 3 mod 7.

But wait, let me check that again. If pages = 2n, then 2n ≡ 3 mod 7. So, solving for n here. Let me write that as an equation: 2n ≡ 3 mod 7. To solve for n, we can multiply both sides by the modular inverse of 2 mod 7. The inverse of 2 mod 7 is a number x such that 2x ≡ 1 mod 7. Let me calculate that. Testing numbers:

2*1=2≡2

2*2=4≡4

2*3=6≡6

2*4=8≡1 mod7. Ah, so 4 is the inverse of 2 mod7.

Therefore, multiplying both sides by 4: n ≡ 4*3 mod7 => n ≡ 12 mod7 => 12 divided by 7 is 1 with remainder 5, so n ≡5 mod7.

Therefore, the number of leaves n must be congruent to 5 mod7, and n >100. We need the smallest such n.

So, the question becomes: find the smallest integer n >100 such that n ≡5 mod7.

So, let's think. The numbers congruent to 5 mod7 are numbers like 5, 12, 19, 26,... each time adding 7. So, starting from 5, add 7 repeatedly.

We need the smallest one greater than 100. Let me find the first n >100 where n ≡5 mod7.

Let me compute 100 divided by 7 to find where 100 falls in the sequence.

7*14=98, 7*14+2=100. So, 100 is 7*14 + 2. So, the next number congruent to 5 mod7 after 100 would be?

Let me think. Let me find the smallest number greater than 100 which is 5 mod7.

Let me compute 100 mod7. Since 7*14=98, 100-98=2. Therefore, 100 ≡2 mod7. So we need to find n such that n ≡5 mod7, and n>100.

So, starting from 100, we can add (5 - 2) mod7? Wait, maybe another way. Let me think.

Let me find how much I need to add to 100 to get to the next number that is 5 mod7.

Since 100 ≡2 mod7. The difference between 5 and 2 is 3, so if I add 3 to 100, then 100 +3=103. But 103 mod7 would be (2 +3)=5 mod7. So 103 ≡5 mod7. Therefore, 103 is the first number after 100 that is congruent to 5 mod7. But wait, is 103 greater than 100? Yes. So is 103 the minimal n?

Wait, but let me check. Let me check what's 7*14=98, then 98+5=103. So 103 is the first number in the sequence 5,12,...,103,... that is larger than 100. Therefore, n=103.

But hold on, is 103 the minimal? Let's check numbers between 100 and 103. For example, 101: 101-98=3, so 3 mod7. 102: 102-98=4 mod7. 103: 103-98=5 mod7. So yes, 103 is the first one. Therefore, the minimal number of leaves is 103.

But wait, let me check if that satisfies the original condition. The number of pages is 2*103=206. Is 206 3 more than a multiple of 7?

Let me compute 206 divided by 7. 7*29=203, 203+3=206. So yes, 206=7*29 +3. Therefore, 206 is indeed 3 more than a multiple of 7. Therefore, 103 leaves would make 206 pages, which satisfies the condition, and 103 is greater than 100.

But the question is, is 103 the minimal? Let me check if there is a number between 100 and 103 that is congruent to 5 mod7. Since 100 ≡2 mod7, then numbers after 100:

100 ≡2

101≡3

102≡4

103≡5

So, yes, 103 is the first one. Therefore, 103 is the minimal number of leaves.

But wait, the problem says "the number of leaves is greater than 100". So leaves must be greater than 100, so n>100. Therefore, 103 is indeed the answer. So why is this question considered challenging? Maybe I made a mistake?

Wait, let me check again. The number of pages is twice the number of leaves, so if leaves is n, pages is 2n. The problem says pages is 3 more than a multiple of 7. So 2n ≡3 mod7.

Earlier, I solved for n ≡ (3 * inverse of 2) mod7. Since inverse of 2 mod7 is 4, so 3*4=12≡5 mod7. Therefore, n≡5 mod7. Therefore, n=7k +5, for some integer k.

So n>100, so 7k +5>100 => 7k>95 =>k>95/7≈13.571. So k≥14.

Thus, the minimal k is 14, so n=7*14 +5=98 +5=103. Yep, same answer. So that seems correct.

Wait, but let me check if n=103 is the minimal. If k=14 gives n=103, then k=13 would be 7*13 +5=91 +5=96, which is less than 100. So yes, 103 is the first n>100. Therefore, 103 is the answer.

But let me check once more. Suppose n=103. Pages=206. 206 divided by 7: 7*29=203, 206-203=3. Yes, 206=7*29 +3. Therefore, 3 more than multiple of 7. So that works. And leaves=103>100. So that's the minimal.

But wait, hold on. Maybe there's another way that the pages could be 3 more than multiple of 7? For example, if pages=7m +3. Since pages=2n, then 2n=7m +3. Therefore, 2n ≡3 mod7. Which is what we had before. So solving for n, which gives n≡5 mod7. So same result.

Therefore, seems like 103 is the answer.

But hold on, I need to make sure that n must be integer. Since number of leaves has to be integer. Well, yes, n is integer. So, 103 is integer. So that's okay.

Alternatively, is there a case where maybe the number of leaves is not an integer? No, because leaves are pieces of paper, so they have to be whole numbers. So n must be integer.

Therefore, I think the answer is 103. However, let me check if there is a smaller n>100 that satisfies 2n ≡3 mod7. Suppose someone thought that maybe pages is 3 more than a multiple of 7, so pages=7k +3, and pages=2n. So, 2n=7k +3. Therefore, n=(7k +3)/2. Therefore, for n to be integer, 7k +3 must be even. Therefore, 7k must be odd, since 7k +3 is even. Therefore, 7k is odd. Since 7 is odd, k must be odd. Therefore, k is odd. Therefore, k=2m +1 for some integer m.

Therefore, substituting back, n=(7*(2m +1) +3)/2=(14m +7 +3)/2=(14m +10)/2=7m +5. Therefore, n=7m +5. Which is same as before. Therefore, n must be congruent to 5 mod7. Therefore, the minimal n>100 is 103. Therefore, same conclusion.

Alternatively, perhaps I can think of it as looking for numbers n>100 where n ≡5 mod7. The numbers would be 103, 110, 117,... So 103 is the first one. Therefore, answer is 103.

Wait, but let me check once again with another approach. Let me consider the equation 2n ≡3 mod7. So, solving for n:

Multiply both sides by inverse of 2 mod7 which is 4, so n ≡ 3*4 mod7 => n≡12 mod7 => n≡5 mod7. So same result.

Therefore, n=7k +5. Then, find minimal k such that 7k +5>100. So, 7k>95 =>k>95/7≈13.571. Therefore, k=14. Therefore, n=7*14 +5=98 +5=103. So same answer.

So, seems like 103 is the answer. But maybe the problem is in the definition of a leaf? Wait, the problem says "a book is said to have n leaves if it is composed of n pieces of paper." So, that is, one leaf is one piece of paper. So, each leaf is one sheet, which has two pages. Therefore, number of pages is 2n. So, that seems correct.

But let me check for example, if a book has 1 leaf, that is 1 sheet of paper, so pages is 2. So 2 pages. So that seems correct. So yes, pages=2n.

Therefore, with that, the problem is solved. So, answer is 103. Therefore, I think that the answer is 103. Therefore, the minimal number of leaves is 103.

But to double-check, let me see if there is a possible mistake here. Let me check the value of n=103. Then pages=206. 206 divided by 7 is 29 with remainder 3. 7*29=203, 206-203=3. So yes, 3 more than a multiple of 7. And 103>100. So that works. Therefore, 103 is correct.

But just to be thorough, let me check the next lower number that is 5 mod7. For example, n=103-7=96. But 96 is less than 100, so not allowed. Hence, 103 is the first one above 100. Therefore, answer is 103.

So I think that is correct.

**Final Answer**
The smallest possible number of leaves is \boxed{103}.

To solve this problem, we need to find the **smallest number of leaves** $ n $ such that:

1. The number of **pages** in the book is **3 more than a multiple of 7**, and
2. The number of **leaves** is **greater than 100**.

---

### Step 1: Understand the Relationship Between Leaves and Pages

We are told that:
- The number of **pages** in the book is **twice** the number of **leaves**, i.e.,
  $$
  \text{pages} = 2n
  $$

Also, the number of pages is said to be **3 more than a multiple of 7**, i.e.,
$$
2n \equiv 3 \pmod{7}
$$

---

### Step 2: Solve the Congruence

We are solving:
$$
2n \equiv 3 \pmod{7}
$$

To eliminate the coefficient 2 on the left, we find the **modular inverse** of 2 modulo 7. The inverse of 2 mod 7 is the number $ x $ such that:
$$
2x \equiv 1 \pmod{7}
$$

Testing small integers:
- $ 2 \cdot 4 = 8 \equiv 1 \pmod{7} $

So, inverse of 2 mod 7 is 4. Multiply both sides of the congruence by 4:
$$
4 \cdot (2n) \equiv 4 \cdot 3 \pmod{7}
\Rightarrow 8n \equiv 12 \pmod{7}
$$
$$
8n \equiv 12 \pmod{7} \Rightarrow n \equiv 5 \pmod{7}
$$

So, the number of **leaves** must satisfy:
$$
n \equiv 5 \pmod{7}
$$

---

### Step 3: Find the Smallest $ n > 100 $ That Satisfies This

We now find the **smallest integer** $ n > 100 $ such that:
$$
n \equiv 5 \pmod{7}
$$

We can write $ n = 7k + 5 $, and solve for $ k $ such that:
$$
7k + 5 > 100 \Rightarrow 7k > 95 \Rightarrow k > \frac{95}{7} \approx 13.57
$$

So the smallest integer $ k $ satisfying this is $ k = 14 $. Then:
$$
n = 7 \cdot 14 + 5 = 98 + 5 = 103
$$

---

### Step 4: Verify the Solution

Check the number of pages:
$$
\text{pages} = 2n = 2 \cdot 103 = 206
$$

Check if 206 is 3 more than a multiple of 7:
$$
206 \div 7 = 29 \text{ remainder } 3 \Rightarrow 206 = 7 \cdot 29 + 3
$$

So, 206 is indeed 3 more than a multiple of 7. Also, $ n = 103 > 100 $.

---

### Final Answer

$$
\boxed{103}
$$

Solution: Let $m$ be the least possible number of leaves. Then $2m$ is the least possible number of pages. We know that $2m\equiv 3\pmod 7\implies 8m \equiv 3\cdot 4\pmod 7\implies m\equiv 12\equiv 5\pmod 7$. So $m=5+7a$ for some positive integer $a$. The smallest such number greater than $100$ is $5+7\cdot 14=\boxed{103}$.


Explanation:
Okay, so I need to compute the cross product (2b - a) × (3c + a) given the cross products of a×b, a×c, and b×c. Hmm, let me recall how cross products work with scalar multiplication and vector addition.

First, I remember that the cross product is bilinear, which means it's linear in both arguments. So, I can distribute the cross product over addition and scalar multiplication. Let me write down the expression step by step.

Starting with (2b - a) × (3c + a). Let me expand this using distributive property.

So, expanding the cross product:

(2b - a) × (3c + a) = 2b × 3c + 2b × a - a × 3c - a × a

Simplify each term:

First term: 2b × 3c = 6(b × c)
Second term: 2b × a = 2(b × a)
Third term: -a × 3c = -3(a × c)
Fourth term: -a × a = - (a × a)

Now, I know that the cross product of any vector with itself is zero, so a × a = 0. So the fourth term is zero.

Therefore, the entire expression simplifies to:

6(b × c) + 2(b × a) - 3(a × c)

Now, I need to handle the terms involving b × a. Remember that cross product is anti-commutative, so b × a = -a × b. Let me substitute that in:

So, 2(b × a) becomes 2(-a × b) = -2(a × b)

Therefore, substituting back:

6(b × c) - 2(a × b) - 3(a × c)

So now, the expression is written in terms of the given cross products: a×b, a×c, and b×c.

Given:

a×b = [6, -7, 3]^T

a×c = [4, 7, 2]^T

b×c = [1, -7, 18]^T

So, substituting these into the expression:

6*(b×c) - 2*(a×b) - 3*(a×c)

Now, compute each component separately.

Let me compute each term:

First term: 6*(b×c) = 6*[1, -7, 18]^T = [6*1, 6*(-7), 6*18]^T = [6, -42, 108]^T

Second term: -2*(a×b) = -2*[6, -7, 3]^T = [-12, 14, -6]^T

Third term: -3*(a×c) = -3*[4, 7, 2]^T = [-12, -21, -6]^T

Now, add all three terms together:

Let me add them component-wise.

First component: 6 + (-12) + (-12) = 6 - 12 -12 = 6 - 24 = -18

Second component: -42 + 14 + (-21) = -42 + 14 -21. Let's compute step by step:

-42 +14 = -28; -28 -21 = -49

Third component: 108 + (-6) + (-6) = 108 -6 -6 = 108 -12 = 96

Therefore, putting it all together, the resulting vector is:

[-18, -49, 96]^T

Wait, let me verify the calculations again to make sure I didn't make any arithmetic errors.

First term components:

6*(b×c) = [6, -42, 108] is correct.

Second term: -2*(a×b) = -2*6 = -12, -2*(-7) = +14, -2*3 = -6. Correct.

Third term: -3*(a×c) = -3*4 = -12, -3*7 = -21, -3*2 = -6. Correct.

Adding the first components: 6 -12 -12. 6 -12 is -6, then -6 -12 is -18. Correct.

Second components: -42 +14 is -28, then -28 + (-21) is -49. Correct.

Third components: 108 -6 is 102, then 102 -6 is 96. Correct.

So the final result is [-18, -49, 96]. Therefore, the cross product (2b - a) × (3c + a) equals that vector.

But let me check again if I expanded the original expression correctly. Let me go back step by step.

Original expression: (2b - a) × (3c + a)

Expanding using distributive property:

First, 2b × 3c = 6(b × c)

Then, 2b × a = 2(b × a)

Then, -a × 3c = -3(a × c)

Then, -a × a = 0.

Yes, so the expansion is correct. Then replacing b × a with -a × b. So:

6(b × c) - 2(a × b) - 3(a × c). Correct.

Substituting the given cross products. Then computing each term.

So I think my calculation is correct.

But just to make sure, let me think if there's another way to verify.

Alternatively, maybe we can compute the cross product using the properties of vectors. But since we don't have the actual vectors a, b, c, but only their cross products, perhaps there is no other way. But maybe we can check if the answer makes sense.

Alternatively, let me think if there's a possibility of miscalculating the signs. For example, in the second term, 2(b × a) becomes -2(a × b). Is that correct?

Yes. Because b × a = -a × b, so 2*(b × a) = -2*(a × b). So that term is correct.

Third term is -3(a × c). Correct.

So adding up those three terms:

First term: 6*(b × c) = [6, -42, 108]

Second term: -2*(a × b) = [-12, 14, -6]

Third term: -3*(a × c) = [-12, -21, -6]

Adding them together:

For x-component: 6 -12 -12 = -18.

For y-component: -42 +14 -21 = -42 +14 is -28, -28 -21 is -49.

For z-component: 108 -6 -6 = 96.

Yes, same as before. So seems correct.

Alternatively, maybe I can check if the answer is consistent with some properties? For example, cross product is anti-commutative, so maybe check if the result is perpendicular to both (2b - a) and (3c + a). But since we don't have the actual vectors, it's hard to verify.

Alternatively, maybe compute the magnitude or something else? Not sure. Alternatively, maybe check if I made a mistake in the sign of the second term. Let me think again:

Original expansion:

(2b - a) × (3c + a) = 2b × 3c + 2b × a - a × 3c - a × a.

Yes, that is correct. Because cross product distributes over addition, so:

(2b - a) × (3c + a) = (2b - a) × 3c + (2b - a) × a = 2b × 3c - a × 3c + 2b × a - a × a.

Yes, so that's 6(b × c) - 3(a × c) + 2(b × a) - 0. Then 2(b × a) is -2(a × b). So yes, same as before.

Therefore, I think the answer is correct. Therefore, the final result is the vector (-18, -49, 96).

But just to be thorough, let me recompute each component once again:

First component:

6*(b×c)_x = 6*1 = 6

-2*(a×b)_x = -2*6 = -12

-3*(a×c)_x = -3*4 = -12

Total x-component: 6 -12 -12 = -18

Second component:

6*(b×c)_y = 6*(-7) = -42

-2*(a×b)_y = -2*(-7) = +14

-3*(a×c)_y = -3*7 = -21

Total y-component: -42 +14 -21 = (-42 -21) +14 = -63 +14 = -49. Correct.

Third component:

6*(b×c)_z = 6*18 = 108

-2*(a×b)_z = -2*3 = -6

-3*(a×c)_z = -3*2 = -6

Total z-component: 108 -6 -6 = 96. Correct.

Okay, so yes, all components check out. Therefore, I can be confident that the answer is correct.

Therefore, the final answer is the vector (-18, -49, 96).

**Final Answer**
\boxed{\begin{pmatrix} -18 \\ -49 \\ 96 \end{pmatrix}}

To solve the problem, we are given three cross products:

$$
\mathbf{a} \times \mathbf{b} = \begin{pmatrix} 6 \\ -7 \\ 3 \end{pmatrix}, \quad
\mathbf{a} \times \mathbf{c} = \begin{pmatrix} 4 \\ 7 \\ 2 \end{pmatrix}, \quad
\mathbf{b} \times \mathbf{c} = \begin{pmatrix} 1 \\ -7 \\ 18 \end{pmatrix}.
$$

We are asked to compute:

$$
(2 \mathbf{b} - \mathbf{a}) \times (3 \mathbf{c} + \mathbf{a})
$$

---

### Step 1: Use Bilinearity of the Cross Product

The cross product is **bilinear**, which means:

- It distributes over vector addition
- It scales linearly with scalar multiplication

We'll expand the expression step by step.

$$
(2 \mathbf{b} - \mathbf{a}) \times (3 \mathbf{c} + \mathbf{a}) =
2 \mathbf{b} \times 3 \mathbf{c} + 2 \mathbf{b} \times \mathbf{a} - \mathbf{a} \times 3 \mathbf{c} - \mathbf{a} \times \mathbf{a}
$$

Simplify each term:

- $ 2\mathbf{b} \times 3\mathbf{c} = 6(\mathbf{b} \times \mathbf{c}) $
- $ 2\mathbf{b} \times \mathbf{a} = 2(\mathbf{b} \times \mathbf{a}) $
- $ -\mathbf{a} \times 3\mathbf{c} = -3(\mathbf{a} \times \mathbf{c}) $
- $ -\mathbf{a} \times \mathbf{a} = 0 $, since the cross product of a vector with itself is zero

So the expression simplifies to:

$$
6(\mathbf{b} \times \mathbf{c}) + 2(\mathbf{b} \times \mathbf{a}) - 3(\mathbf{a} \times \mathbf{c})
$$

Now use **anti-commutativity**: $ \mathbf{b} \times \mathbf{a} = -\mathbf{a} \times \mathbf{b} $, so:

$$
\text{Expression becomes: } 6(\mathbf{b} \times \mathbf{c}) - 2(\mathbf{a} \times \mathbf{b}) - 3(\mathbf{a} \times \mathbf{c})
$$

---

### Step 2: Plug in the Given Vectors

Recall the provided vectors:

- $ \mathbf{b} \times \mathbf{c} = \begin{pmatrix} 1 \\ -7 \\ 18 \end{pmatrix} $
- $ \mathbf{a} \times \mathbf{b} = \begin{pmatrix} 6 \\ -7 \\ 3 \end{pmatrix} $
- $ \mathbf{a} \times \mathbf{c} = \begin{pmatrix} 4 \\ 7 \\ 2 \end{pmatrix} $

Now compute each term:

1. $ 6(\mathbf{b} \times \mathbf{c}) = 6 \cdot \begin{pmatrix} 1 \\ -7 \\ 18 \end{pmatrix} = \begin{pmatrix} 6 \\ -42 \\ 108 \end{pmatrix} $

2. $ -2(\mathbf{a} \times \mathbf{b}) = -2 \cdot \begin{pmatrix} 6 \\ -7 \\ 3 \end{pmatrix} = \begin{pmatrix} -12 \\ 14 \\ -6 \end{pmatrix} $

3. $ -3(\mathbf{a} \times \mathbf{c}) = -3 \cdot \begin{pmatrix} 4 \\ 7 \\ 2 \end{pmatrix} = \begin{pmatrix} -12 \\ -21 \\ -6 \end{pmatrix} $

---

### Step 3: Add the Resulting Vectors

Now, add the three vectors component-wise:

- First component: $ 6 - 12 - 12 = -18 $
- Second component: $ -42 + 14 - 21 = -49 $
- Third component: $ 108 - 6 - 6 = 96 $

---

### Final Answer

$$
(2 \mathbf{b} - \mathbf{a}) \times (3 \mathbf{c} + \mathbf{a}) = \boxed{\begin{pmatrix} -18 \\ -49 \\ 96 \end{pmatrix}}
$$

[Assistant]
Explanation:
Okay, so I need to compute the cross product (2b - a) × (3c + a) given the cross products of a×b, a×c, and b×c. Hmm, let me recall how cross products work with scalar multiplication and vector addition.

First, I remember that the cross product is bilinear, which means it's linear in both arguments. So, I can distribute the cross product over addition and scalar multiplication. Let me write down the expression step by step.

Starting with (2b - a) × (3c + a). Let me expand this using distributive property.

So, expanding the cross product:

(2b - a) × (3c + a) = 2b × 3c + 2b × a - a × 3c - a × a

Simplify each term:

First term: 2b × 3c = 6(b × c)
Second term: 2b × a = 2(b × a)
Third term: -a × 3c = -3(a × c)
Fourth term: -a × a = - (a × a)

Now, I know that the cross product of any vector with itself is zero, so a × a = 0. So the fourth term is zero.

Therefore, the entire expression simplifies to:

6(b × c) + 2(b × a) - 3(a × c)

Now, I need to handle the terms involving b × a. Remember that cross product is anti-commutative, so b × a = -a × b. Let me substitute that in:

So, 2(b × a) becomes 2(-a × b) = -2(a × b)

Therefore, substituting back:

6(b × c) - 2(a × b) - 3(a × c)

So now, the expression is written in terms of the given cross products: a×b, a×c, and b×c.

Given:

a×b = [6, -7, 3]^T

a×c = [4, 7, 2]^T

b×c = [1, -7, 18]^T

So, substituting these into the expression:

6*(b×c) - 2*(a×b) - 3*(a×c)

Now, compute each component separately.

Let me compute each term:

First term: 6*(b×c) = 6*[1, -7, 18]^T = [6*1, 6*(-7), 6*18]^T = [6, -42, 108]^T

Second term: -2*(a×b) = -2*[6, -7, 3]^T = [-12, 14, -6]^T

Third term: -3*(a×c) = -3*[4, 7, 2]^T = [-12, -21, -6]^T

Now, add all three terms together:

Let me add them component-wise.

First component: 6 + (-12) + (-12) = 6 - 12 -12 = 6 - 24 = -18

Second component: -42 + 14 + (-21) = -42 + 14 -21. Let's compute step by step:

-42 +14 = -28; -28 -21 = -49

Third component: 108 + (-6) + (-6) = 108 -6 -6 = 108 -12 = 96

Therefore, putting it all together, the resulting vector is:

[-18, -49, 96]^T

Wait, let me verify the calculations again to make sure I didn't make any arithmetic errors.

First term components:

6*(b×c) = [6, -42, 108] is correct.

Second term: -2*(a×b) = -2*6 = -12, -2*(-7) = +14, -2*3 = -6. Correct.

Third term: -3*(a×c) = -3*4 = -12, -3*7 = -21, -3*2 = -6. Correct.

Adding the first components: 6 -12 -12. 6 -12 is -6, then -6 -12 is -18. Correct.

Second components: -42 +14 is -28, then -28 + (-21) is -49. Correct.

Third components: 108 -6 is 102, then 102 -6 is 96. Correct.

So the final result is [-18, -49, 96]. Therefore, the cross product (2b - a) × (3c + a) equals that vector.

But let me check again if I expanded the original expression correctly. Let me go back step by step.

Original expression: (2b - a) × (3c + a)

Expanding using distributive property:

First, 2b × 3c = 6(b × c)

Then, 2b × a = 2(b × a)

Then, -a × 3c = -3(a × c)

Then, -a × a = 0.

Yes, so the expansion is correct. Then replacing b × a with -a × b. So:

6(b × c) - 2(a × b) - 3(a × c). Correct.

Substituting the given cross products. Then computing each term.

So I think my calculation is correct.

But just to make sure, let me think if there's another way to verify.

Alternatively, maybe we can compute the cross product using the properties of vectors. But since we don't have the actual vectors a, b, c, but only their cross products, perhaps there is no other way. But maybe we can check if the answer makes sense.

Alternatively, let me think if there's a possibility of miscalculating the signs. For example, in the second term, 2(b × a) becomes -2(a × b). Is that correct?

Yes. Because b × a = -a × b, so 2*(b × a) = -2*(a × b). So that term is correct.

Third term is -3(a × c). Correct.

So adding up those three terms:

First term: 6*(b × c) = [6, -42, 108]

Second term: -2*(a × b) = [-12, 14, -6]

Third term: -3*(a × c) = [-12, -21, -6]

Adding them together:

For x-component: 6 -12 -12 = -18.

For y-component: -42 +14 -21 = -42 +14 is -28, -28 -21 is -49.

For z-component: 108 -6 -6 = 96.

Yes, same as before. So seems correct.

Alternatively, maybe I can check if the answer is consistent with some properties? For example, cross product is anti-commutative, so maybe check if the result is perpendicular to both (2b - a) and (3c + a). But since we don't have the actual vectors, it's hard to verify.

Alternatively, maybe compute the magnitude or something else? Not sure. Alternatively, maybe check if I made a mistake in the sign of the second term. Let me think again:

Original expansion:

(2b - a) × (3c + a) = 2b × 3c + 2b × a - a × 3c - a × a.

Yes, that is correct. Because cross product distributes over addition, so:

(2b - a) × (3c + a) = (2b - a) × 3c + (2b - a) × a = 2b × 3c - a × 3c + 2b × a - a × a.

Yes, so that's 6(b × c) - 3(a × c) + 2(b × a) - 0. Then 2(b × a) is -2(a × b). So yes, same as before.

Therefore, I think the answer is correct. Therefore, the final result is the vector (-18, -49, 96).

But just to be thorough, let me recompute each component once again:

First component:

6*(b×c)_x = 6*1 = 6

-2*(a×b)_x = -2*6 = -12

-3*(a×c)_x = -3*4 = -12

Total x-component: 6 -12 -12 = -18

Second component:

6*(b×c)_y = 6*(-7) = -42

-2*(a×b)_y = -2*(-7) = +14

-3*(a×c)_y = -3*7 = -21

Total y-component: -42 +14 -21 = (-42 -21) +14 = -63 +14 = -49. Correct.

Third component:

6*(b×c)_z = 6*18 = 108

-2*(a×b)_z = -2*3 = -6

-3*(a×c)_z = -3*2 = -6

Total z-component: 108 -6 -6 = 96. Correct.

Okay, so yes, all components check out. Therefore, I can be confident that the answer is correct.

Therefore, the final answer is the vector (-18, -49, 96).

**Final Answer**
\boxed{\begin{pmatrix} -18 \\ -49 \\ 96 \end{pmatrix}}

To solve the problem, we are given three cross products:

$$
\mathbf{a} \times \mathbf{b} = \begin{pmatrix} 6 \\ -7 \\ 3 \end{pmatrix}, \quad
\mathbf{a} \times \mathbf{c} = \begin{pmatrix} 4 \\ 7 \\ 2 \end{pmatrix}, \quad
\mathbf{b} \times \mathbf{c} = \begin{pmatrix} 1 \\ -7 \\ 18 \end{pmatrix}.
$$

We are asked to compute:

$$
(2 \mathbf{b} - \mathbf{a}) \times (3 \mathbf{c} + \mathbf{a})
$$

---

### Step 1: Use Bilinearity of the Cross Product

The cross product is **bilinear**, which means:

- It distributes over vector addition
- It scales linearly with scalar multiplication

We'll expand the expression step by step.

$$
(2 \mathbf{b} - \mathbf{a}) \times (3 \mathbf{c} + \mathbf{a}) =
2 \mathbf{b} \times 3 \mathbf{c} + 2 \mathbf{b} \times \mathbf{a} - \mathbf{a} \times 3 \mathbf{c} - \mathbf{a} \times \mathbf{a}
$$

Simplify each term:

- $ 2\mathbf{b} \times 3\mathbf{c} = 6(\mathbf{b} \times \mathbf{c}) $
- $ 2\mathbf{b} \times \mathbf{a} = 2(\mathbf{b} \times \mathbf{a}) $
- $ -\mathbf{a} \times 3\mathbf{c} = -3(\mathbf{a} \times \mathbf{c}) $
- $ -\mathbf{a} \times \mathbf{a} = 0 $, since the cross product of a vector with itself is zero

So the expression simplifies to:

$$
6(\mathbf{b} \times \mathbf{c}) + 2(\mathbf{b} \times \mathbf{a}) - 3(\mathbf{a} \times \mathbf{c})
$$

Now use **anti-commutativity**: $ \mathbf{b} \times \mathbf{a} = -\mathbf{a} \times \mathbf{b} $, so:

$$
\text{Expression becomes: } 6(\mathbf{b} \times \mathbf{c}) - 2(\mathbf{a} \times \mathbf{b}) - 3(\mathbf{a} \times \mathbf{c})
$$

---

### Step 2: Plug in the Given Vectors

Recall the provided vectors:

- $ \mathbf{b} \times \mathbf{c} = \begin{pmatrix} 1 \\ -7 \\ 18 \end{pmatrix} $
- $ \mathbf{a} \times \mathbf{b} = \begin{pmatrix} 6 \\ -7 \\ 3 \end{pmatrix} $
- $ \mathbf{a} \times \mathbf{c} = \begin{pmatrix} 4 \\ 7 \\ 2 \end{pmatrix} $

Now compute each term:

1. $ 6(\mathbf{b} \times \mathbf{c}) = 6 \cdot \begin{pmatrix} 1 \\ -7 \\ 18 \end{pmatrix} = \begin{pmatrix} 6 \\ -42 \\ 108 \end{pmatrix} $

2. $ -2(\mathbf{a} \times \mathbf{b}) = -2 \cdot \begin{pmatrix} 6 \\ -7 \\ 3 \end{pmatrix} = \begin{pmatrix} -12 \\ 14 \\ -6 \end{pmatrix} $

3. $ -3(\mathbf{a} \times \mathbf{c}) = -3 \cdot \begin{pmatrix} 4 \\ 7 \\ 2 \end{pmatrix} = \begin{pmatrix} -12 \\ -21 \\ -6 \end{pmatrix} $

---

### Step 3: Add the Resulting Vectors

Now, add the three vectors component-wise:

- First component: $ 6 - 12 - 12 = -18 $
- Second component: $ -42 + 14 - 21 = -49 $
- Third component: $ 108 - 6 - 6 = 96 $

---

### Final Answer

$$
(2 \mathbf{b} - \mathbf{a}) \times (3 \mathbf{c} + \mathbf{a}) = \boxed{\begin{pmatrix} -18 \\ -49 \\ 96 \end{pmatrix}}
$$

Solution:
Expanding, we get
\begin{align*}
(2 \mathbf{b} - \mathbf{a}) \times (3 \mathbf{c} + \mathbf{a}) &= 6 \mathbf{b} \times \mathbf{c} + 2 \mathbf{b} \times \mathbf{a} - 3 \mathbf{a} \times \mathbf{c} - \mathbf{a} \times \mathbf{a} \\
&= 6 \mathbf{b} \times \mathbf{c} - 2 \mathbf{a} \times \mathbf{b} - 3 \mathbf{a} \times \mathbf{c} - \mathbf{0} \\
&= 6 \begin{pmatrix} 1 \\ - 7 \\ 18 \end{pmatrix} - 2 \begin{pmatrix} 6 \\ - 7 \\ 3 \end{pmatrix} - 3 \begin{pmatrix} 4 \\ 7 \\ 2 \end{pmatrix} \\
&= \boxed{\begin{pmatrix} -18 \\ -49 \\ 96 \end{pmatrix}}.
\end{align*}


Explanation:
Okay, so I need to find how many integer coordinates are strictly inside this rectangle with vertices at (5,4), (5,-4), (-5,4), and (-5,-4). Let me visualize this first. The rectangle is centered at the origin, right? The x-coordinates go from -5 to 5, and the y-coordinates go from -4 to 4. But the question says strictly inside, so I can't include the boundary points.

First, let me recall that the number of integer points inside a rectangle can be calculated using the formula related to the area, but adjusted for the grid. However, since it's strictly inside, maybe I need to subtract the boundary points? Or maybe there's a formula for that. Wait, actually, I remember something about Pick's theorem. Let me think. Pick's theorem says that for a simple polygon with integer coordinates, the area is equal to I + B/2 - 1, where I is the number of interior integer points and B is the number of boundary points. But here, the problem is asking for I, the interior points. But I need to calculate the area first, then figure out B, or maybe it's easier to calculate directly?

Alternatively, maybe I can just count them. Let me check the coordinates. The rectangle spans from x = -5 to x = 5 and y = -4 to y = 4. But strictly inside means that the x-coordinates must be greater than -5 and less than 5, and the y-coordinates must be greater than -4 and less than 4. So for integer coordinates, the x-values can be -4, -3, -2, -1, 0, 1, 2, 3, 4. Similarly, the y-values can be -3, -2, -1, 0, 1, 2, 3. Let me verify that.

For x: the original rectangle has x from -5 to 5 inclusive. But strictly inside would be x from -5 < x < 5. Since x must be integer, that means x can't be -5 or 5, so possible x values are integers from -4 up to 4. Let me count how many that is. From -4 to 4 inclusive, that's 4 - (-4) + 1 = 9? Wait, no. Wait, the formula for the number of integers from a to b inclusive is b - a + 1. So here, from -4 to 4, that's 4 - (-4) + 1 = 4 + 4 + 1 = 9. So 9 values for x.

Similarly for y: original y ranges from -4 to 4 inclusive. Strictly inside would be y from -4 < y < 4, so integers from -3 up to 3. Similarly, number of integers is 3 - (-3) + 1 = 3 + 3 + 1 = 7. So 7 values for y.

Therefore, the total number of integer coordinates strictly inside the rectangle should be 9 * 7 = 63. Is that correct? Wait, but let me think again. Maybe I made a mistake here. Because when I count the number of integer points inside, it's not just the product of the number of x and y values? Wait, actually, if the coordinates are independent, then yes, the total number is the product. For each x in the range, there are 7 y's. So 9*7=63. But maybe I need to check with another method?

Alternatively, using Pick's theorem. Let me try that. First, compute the area of the rectangle. The length along the x-axis is from -5 to 5, so that's 10 units. The length along the y-axis is from -4 to 4, so that's 8 units. So the area is 10*8 = 80.

Now, using Pick's theorem: Area = I + B/2 - 1. So I need to find B, the number of boundary points. Then solve for I. But how do I compute B? The boundary of the rectangle consists of four sides. Each side is a line segment.

For each side, the number of integer points on the segment can be calculated using the formula that for a segment between two points (x1,y1) and (x2,y2), the number of integer points is gcd(delta_x, delta_y) + 1, where delta_x is the difference in x-coordinates and delta_y is the difference in y-coordinates. However, since we want the number of boundary points, we need to count all points on the four sides, but we have to be careful not to double count the vertices.

First, let's compute the number of integer points on each side.

Starting with the right side: from (5,4) to (5,-4). This is a vertical line segment. The delta_x is 0, delta_y is -8. The gcd(0,8) is 8. Therefore, the number of integer points on this side is 8 + 1 = 9. But this includes the endpoints (5,4) and (5,-4). Similarly, the left side: from (-5,4) to (-5,-4). Same calculation: gcd(0,8)=8, so 9 points.

Now the top side: from (-5,4) to (5,4). This is horizontal, delta_x is 10, delta_y is 0. gcd(10,0)=10, so the number of integer points is 10 + 1 = 11. Similarly, the bottom side: from (-5,-4) to (5,-4), same as the top, so 11 points.

So total boundary points would be (9 + 9 + 11 + 11) = 40. But this counts each vertex twice, once for each adjacent side. There are 4 vertices, so we need to subtract 4 to get the correct count. So total B = 40 - 4 = 36.

Now applying Pick's theorem: Area = I + B/2 - 1 => 80 = I + 36/2 -1 => 80 = I + 18 -1 => 80 = I + 17 => I = 80 - 17 = 63. So the number of interior points is 63. Which matches my initial calculation of 9*7=63.

So both methods give me 63. Therefore, I think this is the correct answer. But let me check again my first approach.

So for strictly inside, x ranges from -4 to 4 inclusive. That's 9 values. y ranges from -3 to 3 inclusive. That's 7 values. So 9*7=63. That seems right. For example, for x from -4 to 4 inclusive, that's 9 integers. For y from -3 to 3 inclusive, that's 7 integers. So multiplying gives the total number of interior points. That seems straightforward.

Wait, but why does Pick's theorem give the same answer? Because Pick's theorem is about counting interior points, so both approaches should agree. So my answer is 63.

But let me double-check the first method again. If the rectangle is from -5 to 5 in x, but not including the boundaries, so strictly inside. So for x-coordinates: the original rectangle includes x=-5 and x=5, but we exclude them. So the integer x's that are strictly inside are -4, -3, ..., 3, 4. Wait, hold on: if the rectangle goes from x=-5 to x=5 inclusive, then the interior points have x strictly between -5 and 5. So for integer x, that would be x=-4, -3, -2, -1, 0, 1, 2, 3, 4. Which is indeed 9 values. Similarly for y: the original rectangle has y from -4 to 4 inclusive. Strictly inside is y between -4 and 4, not including the endpoints. So integer y-values are -3, -2, -1, 0, 1, 2, 3. So 7 values. Therefore, total points are 9*7=63. So that's straightforward.

Therefore, I think the answer is 63. But let me check if there is any mistake in considering the coordinates. For example, sometimes people might make a mistake in counting the number of integers in an interval. For example, between -4 and 4 inclusive, how many integers? Let me count: -4, -3, -2, -1, 0, 1, 2, 3, 4. That's 9 numbers, which is correct. Similarly for y: between -3 and 3 inclusive: -3, -2, -1, 0, 1, 2, 3: 7 numbers.

Alternatively, if I use the formula for number of integers between a and b inclusive: b - a + 1. For x: 4 - (-4) +1 = 4 + 4 +1=9. For y: 3 - (-3) +1= 3 + 3 +1=7. So yes, that works.

Therefore, multiplying gives 63. Therefore, I think the answer is 63. However, just to be thorough, let me check with Pick's theorem again.

Earlier, I computed the boundary points as 36. Let me verify that again. Each side: right and left have 9 points each, top and bottom have 11 points each. Total before subtracting overlapping vertices: 9+9+11+11=40. But each vertex is counted twice, once for each adjacent side. There are 4 vertices, so subtract 4. So 40-4=36.

Therefore, B=36. Then, Area is 80. Then, using Pick's theorem: I = Area - B/2 +1. Wait, original formula is Area = I + B/2 -1, so solving for I: I = Area - B/2 +1. So 80 - 36/2 +1 = 80 -18 +1=63. Correct.

So both methods confirm that the answer is 63.

Alternatively, maybe someone might think that the answer is different. Let me think if there is a possibility of miscalculating the area? The rectangle has length 10 (from -5 to 5 is 10 units) and height 8 (from -4 to 4 is 8 units). So area is 10*8=80. Correct.

Another way to think about it: the number of interior points is (length - 2)*(height - 2). Wait, no. Wait, if you have a grid, and you want the number of points strictly inside, it's (number of columns - 2)*(number of rows - 2). Wait, but is that applicable here?

Wait, suppose we consider the rectangle as a grid. If the rectangle spans from x=-5 to x=5, that's 10 units in width. If we think of the grid points, the number of vertical lines is 11 (from x=-5 to x=5). Similarly, horizontal lines: 9 (from y=-4 to y=4). But the number of interior points would be (number of vertical lines - 2)*(number of horizontal lines - 2). Wait, no, maybe not. Wait, actually, the number of interior points would be (number of interior columns) multiplied by (number of interior rows).

But how is that? Let me think. The number of interior columns (i.e., x-values between -5 and 5 exclusive) is from -4 to 4, which is 9 columns. Similarly, interior rows is from -3 to 3, which is 7 rows. So 9*7=63. Which is the same as before.

Alternatively, if I think of the total number of grid points in the rectangle including boundaries, that would be (number of x-values) * (number of y-values). The number of x-values from -5 to 5 inclusive is 11 (from -5 to 5 is 11 numbers). Similarly, number of y-values from -4 to 4 inclusive is 9. So total grid points including boundary is 11*9=99. Then, the number of boundary points is 36 as before. Then number of interior points is total grid points minus boundary points minus the points on the edges? Wait, no. Wait, Pick's theorem says that the total points (including boundary) is I + B, where I is interior and B is boundary? Wait no, actually, in Pick's theorem, the formula is Area = I + B/2 - 1, but the total number of lattice points (including boundary) is I + B. But in our case, we can calculate the total number of lattice points inside and on the boundary as 11*9=99. Therefore, if we have B=36 and I=63, then total is 63+36=99. Which matches. Therefore, that's correct.

Therefore, all methods confirm that the answer is 63.

Wait, but maybe I should check with a smaller rectangle to make sure. For example, let's take a rectangle from (0,0) to (2,2). Then the interior points strictly inside would be (1,1). So 1 point. Let's apply the formula.

Number of interior x-values: from 1 to 1 (since original x from 0 to 2, strictly inside is 1). Similarly y: 1. So 1*1=1. Correct. Using Pick's theorem: area is 4. B: each side. For the rectangle from (0,0) to (2,2). Each side: right side from (2,0) to (2,2): gcd(0,2)=2, so 3 points. Similarly left side: same. Top and bottom: gcd(2,0)=2, so 3 points each. Total before subtracting vertices: 3+3+3+3=12. Subtract 4 for the vertices, so B=8. Then I = Area - B/2 + 1 = 4 - 4 +1 =1. Correct.

Alternatively, total grid points including boundary: (3x3)=9. Then B=8, I=1. Therefore 1+8=9. Correct.

Another check: rectangle from (0,0) to (1,1). The interior points strictly inside would be none. Using formula: interior x: from 0 to 1 exclusive, so no integer points. Hence 0. Using Pick's theorem: area=1. B: each side: for example, right side (1,0) to (1,1): gcd(0,1)=1, so 2 points. Similarly for all sides. Total before subtracting vertices: 2*4=8. Subtract 4: B=4. Then I = 1 - 4/2 +1= 1 -2 +1=0. Correct.

Therefore, the formula works. So applying it to our original problem gives 63, which seems correct.

So I think the answer is 63. Therefore, I can confidently put that as the final answer.

**Final Answer**
\boxed{63}

To determine how many **integer coordinates** are **strictly inside the rectangle** with vertices at $(5,4)$, $(5,-4)$, $(-5,4)$, and $(-5,-4)$, we approach the problem step by step using both **logical counting** and **Pick's Theorem** to verify the result.

---

### **Step 1: Understand the Rectangle's Dimensions**
The rectangle spans:

- **Along the x-axis**: from $x = -5$ to $x = 5$
- **Along the y-axis**: from $y = -4$ to $y = 4$

However, we are asked for **integer coordinates strictly inside** the rectangle, **not including the boundary points**.

This means:

- **Strictly inside x-values** are those strictly greater than $-5$ and less than $5$
  - So, valid integer $x$-values: $-4, -3, -2, -1, 0, 1, 2, 3, 4$
  - That’s 9 values

- **Strictly inside y-values** are those strictly greater than $-4$ and less than $4$
  - So, valid integer $y$-values: $-3, -2, -1, 0, 1, 2, 3$
  - That’s 7 values

---

### **Step 2: Count the Interior Points**
Since the $x$-values and $y$-values are independent, the total number of strictly interior integer coordinates is:

$$
9 \times 7 = 63
$$

This is the basic, straightforward way to count the interior points.

---

### **Step 3: Verify with Pick's Theorem**
To confirm, we apply **Pick's Theorem**, which for a lattice polygon (with integer-coordinate vertices) states:

$$
A = I + \frac{B}{2} - 1
$$

Where:
- $A$ is the area of the polygon,
- $I$ is the number of **interior** lattice points,
- $B$ is the number of **boundary** lattice points (including vertices and edge points).

#### **1. Compute Area**
The rectangle has width: $5 - (-5) = 10$, and height: $4 - (-4) = 8$

So the area is:
$$
A = 10 \times 8 = 80
$$

#### **2. Compute Boundary Points $B$**
We use the **gcd-based formula** for the number of lattice points on each edge of a rectangle.

Each edge is a line segment between two points. The number of lattice points on a segment is:

$$
\text{Number of lattice points} = \gcd(\Delta x, \Delta y) + 1
$$

- **Right side**: $ (5,4)$ to $(5,-4)$ → $\Delta y = -8$, $\gcd(0, 8) = 8$, so 9 points
- **Left side**: $(-5,4)$ to $(-5,-4)$ → same as above → 9 points
- **Top side**: $(-5,4)$ to $(5,4)$ → $\Delta x = 10$, $\gcd(10, 0) = 10$, so 11 points
- **Bottom side**: $(-5,-4)$ to $(5,-4)$ → same as above → 11 points

Total boundary points (before subtracting overlaps):
$$
9 + 9 + 11 + 11 = 40
$$

However, we have **overcounted** the four vertices (each counted twice), so we subtract 4:

$$
B = 40 - 4 = 36
$$

#### **3. Apply Pick’s Theorem**

$$
A = I + \frac{B}{2} - 1 \\
80 = I + \frac{36}{2} - 1 = I + 18 - 1 = I + 17 \\
\Rightarrow I = 80 - 17 = 63
$$

---

### **Conclusion**

Using both direct counting and Pick's Theorem, we arrive at the same result.

$$
\boxed{63}
$$

[Assistant]
Explanation:
Okay, so I need to find how many integer coordinates are strictly inside this rectangle with vertices at (5,4), (5,-4), (-5,4), and (-5,-4). Let me visualize this first. The rectangle is centered at the origin, right? The x-coordinates go from -5 to 5, and the y-coordinates go from -4 to 4. But the question says strictly inside, so I can't include the boundary points.

First, let me recall that the number of integer points inside a rectangle can be calculated using the formula related to the area, but adjusted for the grid. However, since it's strictly inside, maybe I need to subtract the boundary points? Or maybe there's a formula for that. Wait, actually, I remember something about Pick's theorem. Let me think. Pick's theorem says that for a simple polygon with integer coordinates, the area is equal to I + B/2 - 1, where I is the number of interior integer points and B is the number of boundary points. But here, the problem is asking for I, the interior points. But I need to calculate the area first, then figure out B, or maybe it's easier to calculate directly?

Alternatively, maybe I can just count them. Let me check the coordinates. The rectangle spans from x = -5 to x = 5 and y = -4 to y = 4. But strictly inside means that the x-coordinates must be greater than -5 and less than 5, and the y-coordinates must be greater than -4 and less than 4. So for integer coordinates, the x-values can be -4, -3, -2, -1, 0, 1, 2, 3, 4. Similarly, the y-values can be -3, -2, -1, 0, 1, 2, 3. Let me verify that.

For x: the original rectangle has x from -5 to 5 inclusive. But strictly inside would be x from -5 < x < 5. Since x must be integer, that means x can't be -5 or 5, so possible x values are integers from -4 up to 4. Let me count how many that is. From -4 to 4 inclusive, that's 4 - (-4) + 1 = 9? Wait, no. Wait, the formula for the number of integers from a to b inclusive is b - a + 1. So here, from -4 to 4, that's 4 - (-4) + 1 = 4 + 4 + 1 = 9. So 9 values for x.

Similarly for y: original y ranges from -4 to 4 inclusive. Strictly inside would be y from -4 < y < 4, so integers from -3 up to 3. Similarly, number of integers is 3 - (-3) + 1 = 3 + 3 + 1 = 7. So 7 values for y.

Therefore, the total number of integer coordinates strictly inside the rectangle should be 9 * 7 = 63. Is that correct? Wait, but let me think again. Maybe I made a mistake here. Because when I count the number of integer points inside, it's not just the product of the number of x and y values? Wait, actually, if the coordinates are independent, then yes, the total number is the product. For each x in the range, there are 7 y's. So 9*7=63. But maybe I need to check with another method?

Alternatively, using Pick's theorem. Let me try that. First, compute the area of the rectangle. The length along the x-axis is from -5 to 5, so that's 10 units. The length along the y-axis is from -4 to 4, so that's 8 units. So the area is 10*8 = 80.

Now, using Pick's theorem: Area = I + B/2 - 1. So I need to find B, the number of boundary points. Then solve for I. But how do I compute B? The boundary of the rectangle consists of four sides. Each side is a line segment.

For each side, the number of integer points on the segment can be calculated using the formula that for a segment between two points (x1,y1) and (x2,y2), the number of integer points is gcd(delta_x, delta_y) + 1, where delta_x is the difference in x-coordinates and delta_y is the difference in y-coordinates. However, since we want the number of boundary points, we need to count all points on the four sides, but we have to be careful not to double count the vertices.

First, let's compute the number of integer points on each side.

Starting with the right side: from (5,4) to (5,-4). This is a vertical line segment. The delta_x is 0, delta_y is -8. The gcd(0,8) is 8. Therefore, the number of integer points on this side is 8 + 1 = 9. But this includes the endpoints (5,4) and (5,-4). Similarly, the left side: from (-5,4) to (-5,-4). Same calculation: gcd(0,8)=8, so 9 points.

Now the top side: from (-5,4) to (5,4). This is horizontal, delta_x is 10, delta_y is 0. gcd(10,0)=10, so the number of integer points is 10 + 1 = 11. Similarly, the bottom side: from (-5,-4) to (5,-4), same as the top, so 11 points.

So total boundary points would be (9 + 9 + 11 + 11) = 40. But this counts each vertex twice, once for each adjacent side. There are 4 vertices, so we need to subtract 4 to get the correct count. So total B = 40 - 4 = 36.

Now applying Pick's theorem: Area = I + B/2 - 1 => 80 = I + 36/2 -1 => 80 = I + 18 -1 => 80 = I + 17 => I = 80 - 17 = 63. So the number of interior points is 63. Which matches my initial calculation of 9*7=63.

So both methods give me 63. Therefore, I think this is the correct answer. But let me check again my first approach.

So for strictly inside, x ranges from -4 to 4 inclusive. That's 9 values. y ranges from -3 to 3 inclusive. That's 7 values. So 9*7=63. That seems right. For example, for x from -4 to 4 inclusive, that's 9 integers. For y from -3 to 3 inclusive, that's 7 integers. So multiplying gives the total number of interior points. That seems straightforward.

Wait, but why does Pick's theorem give the same answer? Because Pick's theorem is about counting interior points, so both approaches should agree. So my answer is 63.

But let me double-check the first method again. If the rectangle is from -5 to 5 in x, but not including the boundaries, so strictly inside. So for x-coordinates: the original rectangle includes x=-5 and x=5, but we exclude them. So the integer x's that are strictly inside are -4, -3, ..., 3, 4. Wait, hold on: if the rectangle goes from x=-5 to x=5 inclusive, then the interior points have x strictly between -5 and 5. So for integer x, that would be x=-4, -3, -2, -1, 0, 1, 2, 3, 4. Which is indeed 9 values. Similarly for y: the original rectangle has y from -4 to 4 inclusive. Strictly inside is y between -4 and 4, not including the endpoints. So integer y-values are -3, -2, -1, 0, 1, 2, 3. So 7 values. Therefore, total points are 9*7=63. So that's straightforward.

Therefore, I think the answer is 63. But let me check if there is any mistake in considering the coordinates. For example, sometimes people might make a mistake in counting the number of integers in an interval. For example, between -4 and 4 inclusive, how many integers? Let me count: -4, -3, -2, -1, 0, 1, 2, 3, 4. That's 9 numbers, which is correct. Similarly for y: between -3 and 3 inclusive: -3, -2, -1, 0, 1, 2, 3: 7 numbers.

Alternatively, if I use the formula for number of integers between a and b inclusive: b - a + 1. For x: 4 - (-4) +1 = 4 + 4 +1=9. For y: 3 - (-3) +1= 3 + 3 +1=7. So yes, that works.

Therefore, multiplying gives 63. Therefore, I think the answer is 63. However, just to be thorough, let me check with Pick's theorem again.

Earlier, I computed the boundary points as 36. Let me verify that again. Each side: right and left have 9 points each, top and bottom have 11 points each. Total before subtracting overlapping vertices: 9+9+11+11=40. But each vertex is counted twice, once for each adjacent side. There are 4 vertices, so subtract 4. So 40-4=36.

Therefore, B=36. Then, Area is 80. Then, using Pick's theorem: I = Area - B/2 +1. Wait, original formula is Area = I + B/2 -1, so solving for I: I = Area - B/2 +1. So 80 - 36/2 +1 = 80 -18 +1=63. Correct.

So both methods confirm that the answer is 63.

Alternatively, maybe someone might think that the answer is different. Let me think if there is a possibility of miscalculating the area? The rectangle has length 10 (from -5 to 5 is 10 units) and height 8 (from -4 to 4 is 8 units). So area is 10*8=80. Correct.

Another way to think about it: the number of interior points is (length - 2)*(height - 2). Wait, no. Wait, if you have a grid, and you want the number of points strictly inside, it's (number of columns - 2)*(number of rows - 2). Wait, but is that applicable here?

Wait, suppose we consider the rectangle as a grid. If the rectangle spans from x=-5 to x=5, that's 10 units in width. If we think of the grid points, the number of vertical lines is 11 (from x=-5 to x=5). Similarly, horizontal lines: 9 (from y=-4 to y=4). But the number of interior points would be (number of vertical lines - 2)*(number of horizontal lines - 2). Wait, no, maybe not. Wait, actually, the number of interior points would be (number of interior columns) multiplied by (number of interior rows).

But how is that? Let me think. The number of interior columns (i.e., x-values between -5 and 5 exclusive) is from -4 to 4, which is 9 columns. Similarly, interior rows is from -3 to 3, which is 7 rows. So 9*7=63. Which is the same as before.

Alternatively, if I think of the total number of grid points in the rectangle including boundaries, that would be (number of x-values) * (number of y-values). The number of x-values from -5 to 5 inclusive is 11 (from -5 to 5 is 11 numbers). Similarly, number of y-values from -4 to 4 inclusive is 9. So total grid points including boundary is 11*9=99. Then, the number of boundary points is 36 as before. Then number of interior points is total grid points minus boundary points minus the points on the edges? Wait, no. Wait, Pick's theorem says that the total points (including boundary) is I + B, where I is interior and B is boundary? Wait no, actually, in Pick's theorem, the formula is Area = I + B/2 - 1, but the total number of lattice points (including boundary) is I + B. But in our case, we can calculate the total number of lattice points inside and on the boundary as 11*9=99. Therefore, if we have B=36 and I=63, then total is 63+36=99. Which matches. Therefore, that's correct.

Therefore, all methods confirm that the answer is 63.

Wait, but maybe I should check with a smaller rectangle to make sure. For example, let's take a rectangle from (0,0) to (2,2). Then the interior points strictly inside would be (1,1). So 1 point. Let's apply the formula.

Number of interior x-values: from 1 to 1 (since original x from 0 to 2, strictly inside is 1). Similarly y: 1. So 1*1=1. Correct. Using Pick's theorem: area is 4. B: each side. For the rectangle from (0,0) to (2,2). Each side: right side from (2,0) to (2,2): gcd(0,2)=2, so 3 points. Similarly left side: same. Top and bottom: gcd(2,0)=2, so 3 points each. Total before subtracting vertices: 3+3+3+3=12. Subtract 4 for the vertices, so B=8. Then I = Area - B/2 + 1 = 4 - 4 +1 =1. Correct.

Alternatively, total grid points including boundary: (3x3)=9. Then B=8, I=1. Therefore 1+8=9. Correct.

Another check: rectangle from (0,0) to (1,1). The interior points strictly inside would be none. Using formula: interior x: from 0 to 1 exclusive, so no integer points. Hence 0. Using Pick's theorem: area=1. B: each side: for example, right side (1,0) to (1,1): gcd(0,1)=1, so 2 points. Similarly for all sides. Total before subtracting vertices: 2*4=8. Subtract 4: B=4. Then I = 1 - 4/2 +1= 1 -2 +1=0. Correct.

Therefore, the formula works. So applying it to our original problem gives 63, which seems correct.

So I think the answer is 63. Therefore, I can confidently put that as the final answer.

**Final Answer**
\boxed{63}

To determine how many **integer coordinates** are **strictly inside the rectangle** with vertices at $(5,4)$, $(5,-4)$, $(-5,4)$, and $(-5,-4)$, we approach the problem step by step using both **logical counting** and **Pick's Theorem** to verify the result.

---

### **Step 1: Understand the Rectangle's Dimensions**
The rectangle spans:

- **Along the x-axis**: from $x = -5$ to $x = 5$
- **Along the y-axis**: from $y = -4$ to $y = 4$

However, we are asked for **integer coordinates strictly inside** the rectangle, **not including the boundary points**.

This means:

- **Strictly inside x-values** are those strictly greater than $-5$ and less than $5$
  - So, valid integer $x$-values: $-4, -3, -2, -1, 0, 1, 2, 3, 4$
  - That’s 9 values

- **Strictly inside y-values** are those strictly greater than $-4$ and less than $4$
  - So, valid integer $y$-values: $-3, -2, -1, 0, 1, 2, 3$
  - That’s 7 values

---

### **Step 2: Count the Interior Points**
Since the $x$-values and $y$-values are independent, the total number of strictly interior integer coordinates is:

$$
9 \times 7 = 63
$$

This is the basic, straightforward way to count the interior points.

---

### **Step 3: Verify with Pick's Theorem**
To confirm, we apply **Pick's Theorem**, which for a lattice polygon (with integer-coordinate vertices) states:

$$
A = I + \frac{B}{2} - 1
$$

Where:
- $A$ is the area of the polygon,
- $I$ is the number of **interior** lattice points,
- $B$ is the number of **boundary** lattice points (including vertices and edge points).

#### **1. Compute Area**
The rectangle has width: $5 - (-5) = 10$, and height: $4 - (-4) = 8$

So the area is:
$$
A = 10 \times 8 = 80
$$

#### **2. Compute Boundary Points $B$**
We use the **gcd-based formula** for the number of lattice points on each edge of a rectangle.

Each edge is a line segment between two points. The number of lattice points on a segment is:

$$
\text{Number of lattice points} = \gcd(\Delta x, \Delta y) + 1
$$

- **Right side**: $ (5,4)$ to $(5,-4)$ → $\Delta y = -8$, $\gcd(0, 8) = 8$, so 9 points
- **Left side**: $(-5,4)$ to $(-5,-4)$ → same as above → 9 points
- **Top side**: $(-5,4)$ to $(5,4)$ → $\Delta x = 10$, $\gcd(10, 0) = 10$, so 11 points
- **Bottom side**: $(-5,-4)$ to $(5,-4)$ → same as above → 11 points

Total boundary points (before subtracting overlaps):
$$
9 + 9 + 11 + 11 = 40
$$

However, we have **overcounted** the four vertices (each counted twice), so we subtract 4:

$$
B = 40 - 4 = 36
$$

#### **3. Apply Pick’s Theorem**

$$
A = I + \frac{B}{2} - 1 \\
80 = I + \frac{36}{2} - 1 = I + 18 - 1 = I + 17 \\
\Rightarrow I = 80 - 17 = 63
$$

---

### **Conclusion**

Using both direct counting and Pick's Theorem, we arrive at the same result.

$$
\boxed{63}
$$

Solution: Points with integer coordinates are called lattice points. The length of the rectangle is $5 - (-5) = 10$ units. There will be 9 lattice positions between the two vertical sides of the rectangle. The height of the rectangle is $4 - (-4) = 8$ units. There will be 7 lattice positions between the top and bottom of the rectangle. That's a total of $9 \times 7 = \boxed{63}$ lattice points.