The Factor's Factors

In order to know how many of 720's divisors 720 are also divisors of one another, we can start by calculating its prime factors:

720 = 2^4 $\times$ 3^2 $\times$ 5^1

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Now, let's see how many divisors it has:

(4+1) $\times$ (2+1) $\times$ (1+1) = 5 $\times$ 3 $\times$ 2 = 30 divisors (the exponents of the prime factors plus 1)

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So, we get all possible values of a and b:

a = 2^(4-x) $\times$ 3^(2-y) $\times$ 5^(1-z), where (4-x), (2-y) and (1-z) are positive or 0 and smaller than 4, 2, and 1 respectively

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If a $\geq$ b, then all of a's divisors are a possible value of b, because they're both divisors of 720, so, for each value of a we get:

(5-x) $\times$ (3-y) $\times$ (2-z) possible values for b!

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Iterating through all possible values for x, y, and z we get 270!

Note: Because I used prime factors, I called the factors of 720 its divisors