Divisibility (Hard)

Divisibility: core idea

• 1. How many integers from $1$ to $500$ are divisible by $6$ or $10$?

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• 2. Find the remainder when $7^{100}$ is divided by $13$.

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• 3. For how many digits $a$ is $3a72$ divisible by $11$?

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• 4. Find the smallest positive multiple of $37$ with only digit $1$.

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• 5. How many divisors of $360$ are divisible by $12$?

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• 6. Find the last two digits of $3^{20}$.

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• 7. Find the least $n>0$ for which $2^n$ is divisible by $32$ but not $64$.

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• 8. Find the remainder of $1+2+\cdots+100$ upon division by $7$.

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• 9. Find the least positive integer congruent to $2\pmod5$ and $3\pmod7$.

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