The positive two-digit integers x and y…
We can solve this question using the natural relationships that all two-digit numbers have. As an example, we can express 37 as (10 x 3) + 7. We multiply the digit in the tens position by 10 and then add the digit in the ones position.
If we let a = the tens digit of x and b = the ones digit of x, we know:
x = 10a + b
Since the digits of y are the reverse of those of x, we can express y as:
y = 10b + a
When we sum x and y we obtain:
x + y = 10a + b + 10b + a = 11a + 11b
x + y = 11(a + b)
The final expression 11(a + b) is a multiple of 11, and therefore 11 divides evenly into it.
We see, therefore, that 11 must be a factor of x + y.