Math Core

This mock contest covers the whole Number Theory unit: divisor formulas, modular arithmetic, linear Diophantine equations, the Euclidean algorithm, bases and digits, Legendre's formula, and floor functions.

Question 1

How many positive divisors does $10!$ have?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 2

What is the sum of the positive divisors of $864 = 2^5 \cdot 3^3$ ?

Question 3

What is the remainder when $3^{2026}$ is divided by $11$ ?

Question 4

What are the last two digits of $13^{42}$ ? (Enter them as a number.)

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 5

How many ordered pairs of positive integers $(x, y)$ satisfy $5x + 8y = 500$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 6

A game awards $8$ points or $13$ points per play. What is the largest total score that is impossible to reach?

Question 7

What is $\gcd(4199, 3553)$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 8

As $n$ ranges over the positive integers, which of the following describes all possible values of $\gcd(3n + 1,\ 5n + 2)$ ?

Question 9

How many digits does $8^{20} \cdot 5^{55}$ have in base ten?

Question 10

Write $2026$ in base $6$ . (Enter the digits as a number.)

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 11

How many zeros does $1000!$ end in when written in base $18$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 12

For how many integers $k$ with $0 \le k \le 200$ is $\dbinom{200}{k}$ odd?

Question 13

Find the positive real number $x$ such that $x \lfloor x \rfloor = 70$ . (Enter a fraction or decimal.)

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 14

What is $\displaystyle\sum_{k=1}^{22} \left\lfloor \frac{5k}{23} \right\rfloor$ ?

Question 15

What are the last three digits of $2^{2026}$ ? (Enter them as a number.)

Enter a number. Fractions like 3/4 and sqrt(2) are OK.