Unit 3 test: Number Theory

Find the remainder when $3^{2026} + 5^{2026}$ is divided by $13$.

Find the last three digits of $2026^{2026}$.

Find the remainder when $3^{3^{27}}$ is divided by $1000$. (Note $3^{3^{27}} = 3^{(3^{27})}$.)

Find the smallest integer $n > 1$ that leaves remainder $1$ when divided by each of $2, 3, 4, 5, 6$ and is divisible by $11$.

Find the smallest integer $n > 1$ such that $n^2 - 1$ is divisible by $1000$.

Find the sum of all primes $p$ for which the decimal expansion of $\dfrac1p$ is purely periodic with period exactly $6$.

For how many integers $a$ with $1 \le a \le 1000$ is $a^{20} - 1$ divisible by $1000$?

How many ordered pairs of positive integers $(x, y)$ satisfy $xy = 5x + 12y + 100$?

Find the largest positive integer that cannot be written as $11a + 13b$ with $a$ and $b$ positive integers.

A positive integer $N$ is written $\underline{a}\,\underline{b}\,\underline{c}$ in base $6$ and $\underline{c}\,\underline{b}\,\underline{a}$ in base $11$, where $a$ and $c$ are nonzero. Find $N$ in base ten.

Find the sum of all positive integers $n \le 200$ that have exactly $10$ positive divisors.

Find the only positive integer $n$ such that $n = 12 \cdot d(n)$, where $d(n)$ is the number of positive divisors of $n$.

Find the largest integer $k$ such that $2^k$ divides $7^{2048} - 1$.

For how many integers $k$ with $0 \le k \le 1000$ is $\dbinom{1000}{k}$ not divisible by $7$?

Find $\displaystyle\sum_{n=1}^{64} v_2\!\left(5^n - 3^n\right)$.