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For (x,y) positive integers, let 10xy + 14x + 15y = 183.  Find x + y.

 Dec 20, 2020
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First, find a value of y that could work.

Suppose y is even. This would mean that the expression is (a multiple of 10) + 14x + (a multiple of 10) = 183.

This cannot work because any integer x times 14 is even, and the result must be odd (183).

Now we know y must be odd. Therefore, if we look at the units digits alone, we will have:

0 + (even) + 5 = 183.

This means that the "even" number (we know it is even because it is a multiple of 14) must have a units digit of 8. 28 ends in an 8, so we could try x = 2. This gives us:

20y + 28 + 15y = 183,

35y = 155.

However, 155 is not divisible by 35, so y would not be an integer.

The next number divisible by 14 that ends in 8 is 98. Substituting x = 7 gives us:

70y + 98 + 35y = 183,

105y = 105, 

y = 1.

All this is is strategic trial and error, but it does the job. We get x = 7, y = 1.

The answer asks for the sum, which is 8.

 

laughlaughlaugh

 Dec 20, 2020

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