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Problem 628

AIME late

Number theory Difficulty 5.1 Find the answer

19. Among the 2015 natural numbers from 1 to 2015, the maximum number of numbers that can be found such that the product of this number and 240 is a perfect square is $\qquad$ _.

Your answer

Official solution

Analysis: $240=2^{4} \times 3 \times 5$ , so the selected number must have at least one 3 and 5 in its prime factorization, meaning this number is a multiple of 15. $2015 \div 15=134 \cdots \cdots 5$ , so the multiples of 15 from $1^{2} \sim 11^{2}$ are all valid, meaning there are 11 such numbers.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.