What is the smallest positive integer $x$ that, when multiplied by $400$, produces a product that is a multiple of $576$?

The greatest common factor of 400 and 576 is 16, so the least common multiple is ...

... 400 × 576 / 16 = 400 × 36

The smallest positive integer x is 36.

Comment on greatest common factor

There are at least a couple of ways to find the greatest common factor (GCG) of 400 and 576. One is to simply factor the numbers.

... 400 = 2⁴×5²

... 576 = 2⁶×3²

The largest factor common to both these numbers is 2⁴ = 16.

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Another way is to use Euclid's algorithm.

... 576/400 = 1 r 176

... 400/176 = 2 r 48

... 176/48 = 3 r 32

... 48/32 = 1 r 16

... 32/16 = 2 r 0 . . . . . so 16 is the GCF

Comment on least common multiple

The least common multiple is the product of unique prime factors to the highest power that they appear in either number. Here, that is ...

... 2⁶×3²×5² = LCM(400, 576)

... = (2²×3²)×400 = 36×400

This can also be found by

... LCM(a, b) = a×b/GCF(a, b) . . . . . which is what we did above

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