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# Number Theory

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If m is the smallest positive integer such that m! is a multiple of 4125, and n is the smallest positive integer such that n! is a multiple of 2816000, then find n-m.

May 14, 2022

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Suppose that m is the smallest positive integer such that m! is a multiple of 4125.

First we need to take a look at what 4125 really is.

$$4125 = 3 \times 5^3 \times 11$$

So, is 11! a multiple of 4125? No, because if you expand 11! = 11 * 10 * 9 * ... * 1, you will notice only 5 and 10 has a factor of 5, so the exponent of 5 in the prime factorization of 11! is 2, not 3. We need another multiple of 5 in the factorial so to make it 3. Therefore, m = 15.

Can you try to find the value of n on your own?

May 14, 2022