Divisibility (Medium)

AMC 10/12 mid (≈Q9–16) — multi-step.

Divisibility: core idea

gcd/lcm.

Find the smallest positive integer divisible by both $12$ and $18$ .

Hint: Use lcm/gcd and count multiples carefully. Answer: 36.

1. How many integers from $1$ to $100$ are divisible by $4$ but not by $8$ ?
2. Find the digit $x$ such that $7x32$ is divisible by $9$ .
3. How many positive integers less than $100$ are divisible by both $6$ and $15$ ?
4. What is the least positive integer leaving remainder $2$ when divided by $5$ and remainder $3$ when divided by $4$ ?
5. Find the number of trailing zeros in $25!$ .
6. A number is divisible by $12$ and $15$ . What is the least possible positive value?
7. How many factors of $360$ are divisible by $6$ ?
8. Find the least positive multiple of $9$ whose digits sum to $18$ .
9. How many multiples of $12$ lie between $100$ and $200$ , inclusive?