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Prime Factorization (Hard)

Late AMC 10/12 through early AIME — dense computation.

EasyMediumHardAIME / USAJMO / AMO

Prime Factorization: core idea

Trailing zeros style mid.

Example problem: hint and answer

Find gcd⁡(263452,243653)\gcd(2^6 3^4 5^2,2^4 3^6 5^3).

Hint: Work with the exponents in the prime factorizations. Answer: 32400.

Practice (9 problems — type your answers)

  • 1. How many positive divisors does 243272^4 3^2 7 have?

  • 2. Find lcm⁡(84,150)\operatorname{lcm}(84,150).

  • 3. Find the least nn such that n!n! is divisible by 28332^8 3^3.

  • 4. How many trailing zeros does 100!100! have?

  • 5. Distinct primes p,qp,q satisfy p2q3=864p^2q^3=864. Find p+qp+q.

  • 6. Find the sum of distinct prime factors of 46204620.

  • 7. What least integer multiplies 233252^3 3^2 5 into a square?

  • 8. How many square divisors has 2735522^7 3^5 5^2?

  • 9. Find the greatest cube divisor of 2837542^8 3^7 5^4.

Prime Factorization Practice Problems — Hard