## August 2023

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In the Math Corner of my August 2023 Diary I posted the following brainteaser that I picked up from Twitter … sorry: from X.

Find all nonnegative integer solutions to the equation 15x+ 10y+ 9z= 71.

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**• Solution**

Too easy! Rewrite the left-hand side as 5(3*x* + 2*y*) + 9*z*. So we're adding some multiple of 5
to some multiple of 9 to get 71.

Available multiples of 9 are 9, 18, 27, 36, 45, 54, 63. They differ from 71 by 62, 53, 44, 35, 26, 17, and 8. The only one of those that's a multiple of 5 is 35; so we're looking at 35 + 36 = 71.

It follows that 3*x* + 2*y* = 7 and *z* = 4. Working similar logic on
3*x* + 2*y* (i.e. available multiples of 2 are 2, 4, and 6; they differ from 7 by 5, 3, and 1; only 3 is a multiple of 3) gets
you *x* = 1 and *y* = 2.

So *x* = 1, *y* = 2, and *z* = 4.

I've gotta stop hanging out on Twi … sorry: on X.