Number of factors

There’s a neat trick for finding the number of factors of a number.  First find the prime factorization of the number, for example:

756 = 2^2 \cdot 3^3 \cdot 7^1

Add +1 to the power of each prime factor and then multiply those numbers together.  In this example,

(2+1)\cdot(3+1)\cdot(1+1)= 3\cdot 4 \cdot 2 = 24 factors.

This reflects the fact that each factor of 756 contains 2^n where n= 0, 1, or 2 and 3^m, where m = 0, 1, 2, or 3 and 7^p where p = 0 or 1.

I used this property to solve a problem in a number theory class I’m taking.  The problem asks to compute the sum of all positive integers k such that 1984k has  21 positive factors.

Since 21 = 3\cdot7 , working backward we are looking for a prime factorization with 2 primes raised to the powers of 2 and 6.  The prime factorization of 1984 = 2^6\cdot 31 so we have one prime raised to the sixth power.  All we need is the other prime to be squared.  Setting k=31 gives us

1984k = 2^6\cdot 31^2 with (6+1)(2+1)= 21 divisors.

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