How many divisors of 33! are perfect squares?

Grab a pencil and paper. Answering the following questions will lead you to the solution. Read the [Hint] if you need it and write down your answer. After you've got it, hit the [Answer] link and see how you did. Good luck along your path of discovery!

  1. What is the prime factorization of 33! ?

    [Hint]      [Answer]
  2. What is interesting about the exponents of prime factors of perfect squares?

  3. What is the prime factorization of the largest perfect square that divides 33! ?

  4. 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: (22)x · (32)y · (52)z · · ·

  5. How many ways can the squared primes be combined? This is the number of perfect square divisors of 33!

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