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(3x + 2y) + 9z. 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 3x + 2y = 7 and z = 4. Working similar logic on 3x + 2y (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.