
How many divisors of 33! are perfect squares?
What is the prime factorization of 33! ?
33! = 2^{31} · 3^{15} ·
5^{7} · 7^{4} · 11^{3} ·
13^{2} · 17 · 19 ·
23 · 29 · 31

What is interesting about the exponents of prime factors of perfect squares?
All of the exponents are even numbers.

What is the prime factorization of the largest perfect square that divides 33! ?
[Answer]
All of the perfect square divisors of 33! are also divisors of the largest
perfect square divisor. Since we are looking for divisors that are perfect
squares it will make things easier to rewrite the previous factorization as
powers of squared primes:
(2^{2})^{x} ·
(3^{2})^{y} ·
(5^{2})^{z} · · ·
How many ways can the squared primes be combined? This is the number of perfect
square divisors of 33!
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