[["Problem: Determine the remainder of 54 (mod 6).\nAnswer:",["\n"]],["Problem: Let $n = 3^{17} + 3^{10}$. It is known that $11$ divides into $n+1$. If $n$ can be written in base $10$ as $ABCACCBAB$, where $A,B,C$ are distinct digits such that $A$ and $C$ are odd and $B$ is not divisible by $3$, find $100A + 10B + C$.\nAnswer:",["\n"]],["Problem: Find $4321_{5}-1234_{5}$. Express your answer in base $5$.\nAnswer:",["\n"]],["Problem: Determine the remainder when $$1+12+123+1234+12345+123456+1234567+12345678$$is divided by $5$.\nAnswer:",["\n"]],["Problem: If $a,b,c$ are positive integers less than $13$ such that \\begin{align*}\n2ab+bc+ca&\\equiv 0\\pmod{13}\\\\\nab+2bc+ca&\\equiv 6abc\\pmod{13}\\\\\nab+bc+2ca&\\equiv 8abc\\pmod {13}\n\\end{align*}then determine the remainder when $a+b+c$ is divided by $13$.\nAnswer:",["\n"]],["Problem: A number is chosen uniformly at random from among the positive integers less than $10^8$. Given that the sum of the digits of the number is 9, what is the probability that the number is prime?\nAnswer:",["\n"]],["Problem: How many ordered pairs of positive integers $(m,n)$ satisfy $\\gcd(m,n) = 2$ and $\\mathop{\\text{lcm}}[m,n] = 108$?\nAnswer:",["\n"]],["Problem: When working modulo $m$, the notation $a^{-1}$ is used to denote the residue $b$ for which $ab\\equiv 1\\pmod{m}$, if any exists. For how many integers $a$ satisfying $0 \\le a < 100$ is it true that $a(a-1)^{-1} \\equiv 4a^{-1} \\pmod{20}$?\nAnswer:",["\n"]],["Problem: When the base 10 integer 269 is converted to base 5, what is the sum of the digits?\nAnswer:",["\n"]],["Problem: What is the sum of all positive integer values of $n$ such that $n^2$ is a factor of $1200$?\nAnswer:",["\n"]]]
