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