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Question

Find the smallest possible integer y such that 288y is a multiple of 5616.Thanks!

Answer

apple

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What?

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288 = 2 x 144
= 2 x 2 x 72
= 2 x 2 x 2 x 36
= 2 x 2 x 2 x 2 x 18
= 2 x 2 x 2 x 2 x 2 x 9
= 2 x 2 x 2 x 2 x 2 x 3 x 3
= 25 x 32

5,616 = 2 x 2,808
= 2 x 2 x 1,404
= 2 x 2 x 2 x 702
= 2 x 2 x 2 x 2 x 351
= 2 x 2 x 2 x 2 x 3 x 117
= 2 x 2 x 2 x 2 x 3 x 3 x 39
= 2 x 2 x 2 x 2 x 3 x 3 x 3 x 13
= 24 x 33 x 13

In order for 288y to be a multiple of 5,616… the prime factors in the final product must be equal to or more than the multiplier.

∴ we just need one more 2

5,616 x 2 = 11,232

288y = 11,232
y = 11,232 ÷ 288
= 39


Check:

288 x 39 = 11,232

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