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# number theory

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If x^3 is a positive factor of 10!*11!, how many possible integer values of x are there?

May 23, 2021

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First, let's find the prime factorization of $$10! \cdot 11!$$

$$10! \cdot 11! = 11 \cdot 10 \cdot10 \cdot9 \cdot9 \cdot8 \cdot8 \cdot7 \cdot7 \cdot6 \cdot6 \cdot5 \cdot5 \cdot4 \cdot4 \cdot3 \cdot3 \cdot2 \cdot2 \cdot1 \cdot1 = 2^{16} \cdot3^{8} \cdot5^{4} \cdot7^{2} \cdot11^{1}$$

Next, let's test values for x (by using the prime factorizion, of course!):

x = 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, ...

I will let you finish the problem. Keep in mind that the highest x can be is $$2^{5}\cdot3^{2}\cdot5=1440$$.

Caution: It is important to note that the question says "integers" instead of "positive integers." The cube of a negative number is still negative, so it will not change the answer to this particular problem. However, if x was squared or raised to any even power, we would have to include the negatives.

May 23, 2021