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2021/22 AHSMC Part 1 Problem 5

Problem 5 of 16

For how many integers nn is the fraction 6300n\dfrac{6300}{n} the square of an integer?

Answer choices

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Solution

Since 6300=22⋅32⋅52⋅76300 = 2^2 \cdot 3^2 \cdot 5^2 \cdot 7, the positive integer nn could take one of the following values: 7,22⋅7,32⋅7,52⋅7,22⋅32⋅7,22⋅52⋅7,32⋅52⋅7,22⋅32⋅52⋅7 7,\quad 2^2 \cdot 7,\quad 3^2 \cdot 7,\quad 5^2 \cdot 7,\quad 2^2 \cdot 3^2 \cdot 7,\quad 2^2 \cdot 5^2 \cdot 7,\quad 3^2 \cdot 5^2 \cdot 7,\quad 2^2 \cdot 3^2 \cdot 5^2 \cdot 7 and hence there are 88 possible values for nn. The answer is (C).

University of Alberta