Unit 3 test: Number Theory

The three-digit number $4A6$ is divisible by $9$. What is the digit $A$?

Every four-digit number of the form $\underline{A}\,\underline{B}\,\underline{B}\,\underline{A}$ (such as $3773$) is divisible by which of the following?

What is the sum of the distinct prime factors of $2310$?

How many zeros are at the end of $125!$?

What is the greatest common divisor of $252$ and $588$?

Two positive integers have a product of $2016$ and a greatest common divisor of $12$. What is their least common multiple?

How many positive divisors does $1440$ have?

What is the smallest positive integer with exactly $10$ positive divisors?

What is the remainder when $2^{100}$ is divided by $9$?

A whole number $n$ leaves remainder $4$ when divided by $7$. What is the remainder when $n^2 + 2n$ is divided by $7$?

What is the units digit of $8^{2026}$?

What is the units digit of $1 \times 3 \times 5 \times 7 \times \dots \times 99$, the product of all odd numbers from $1$ to $99$?

Write the base-ten number $2026$ in base five. (Enter the digits as a number.)

In some base $b$, $52_b + 34_b = 106_b$. What is $b$?