Prime Factorization (AIME / USAJMO / AMO)

Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.

Proof track: treat each problem as a full olympiad writeup on paper. Here you only enter the final answer (integer or short expression).

Prime Factorization: core idea

p-adic orders.

**(Proof track.)** Give a complete justification. Find the smallest square divisible by $2^3 3^4 5$ .

Hint: Factor completely and compare prime exponents. Answer: 32400.

No proof submission required — answers only.

1. **(Proof track.)** Give a complete justification. How many divisors of $2^5 3^3 7^2$ are perfect cubes?
2. **(Proof track.)** Give a complete justification. Find $\gcd(756,990)$ .
3. **(Proof track.)** Give a complete justification. Find $\operatorname{lcm}(72,105)$ .
4. **(Proof track.)** Give a complete justification. Find the exponent of $3$ in $100!$ .
5. **(Proof track.)** Give a complete justification. Find the least $n$ such that $72\mid n!$ .
6. **(Proof track.)** Give a complete justification. Find the sum of all divisors of $2^3 3^2$ .
7. **(Proof track.)** Give a complete justification. How many positive integers divide $10!$ but not $9!$ ?
8. **(Proof track.)** Give a complete justification. Find the least integer with exactly $12$ divisors.
9. **(Proof track.)** Give a complete justification. Find the greatest common divisor of $2^{10}-1$ and $2^6-1$ .