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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.

Mar 30, 2021

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$$4125=3\times 5^3\times 11$$ first 3 multiple of 5 : 5,10,15

m = 15

$$2816000=2^{11}\times 5^3\times 11$$ multiple of 2 : $$2, 2^2, 2\times 3, 2^3, 2\times 5, 2^2\times 3, 2\times 7$$

n=15

n - m = 15 - 15 = 0

Mar 30, 2021