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Problem 252

AMC 10/12, early questions

Number theory Difficulty 4.0 Find the answer Japan 2015 · Japan · 2015

How many non-square numbers are there among the positive factors of the number $6000$ ?

Your answer

Official solution

[34]
Since $6000 = 2^4 3^1 5^3$ , positive factors of $6000$ can be written as $2^a 3^b 5^c$ where $a = 0, 1, 2, 3, 4$ ; $b = 0, 1$ ; $c = 0, 1, 2, 3$ . Therefore, there are altogether $5 \cdot 2 \cdot 4 = 40$ positive factors of $6000$ . Among them, square numbers arise only when all of $a, b, c$ are even, and this happens only $3 \cdot 1 \cdot 2 = 6$ times. Therefore, there are $40 - 6 = 34$ non-square factors of $6000$ .

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