Question

If x, y are integers, and 17 divides both the expressions x² - 2xy + 2y² - 5x + 7y and x² - 3xy + 2y² + x – y, then prove that 17 divides xy – 12x + 15y.

Answer 1

Explanation: 

Observe that x² - 3xy + 2y² + x - y = (x - y)(x - 2y + 1) . Thus 17 divides either x – y or x - 2y + 1. Suppose that 17 divides x - y. In this case x ≡ y (mod 17) and hence 

x² - 2xy + 2y² - 5x + 7y ≡ y² - 2y² + y2 - 5y + 7y≡2y (mod 17) 

Thus the given condition that 17 divides x² - 2xy + y² -5x + 7y implies that 17 also divides 2y and hence y itself. But then x ≡ y (mod 17) implies that 17 divides x also. Hence in this case 17 divides xy – 12x +15y . 

Suppose on the other hand that 17 divides x -2y +1 . Thus x ≡ 2y – 1(mod 17) and hence 

x² - 2xy + y² - 5x + 7y ≡ y 2 - 5y + 6 (mod 17) 

Thus 17 divides y² - 5y + 6 . But x ≡ 2y – 1(mod 17) also implies that xy – 12x + 15y ≡ 2(y² - 5y + 6) (mod 17) 

Since 17 divides (y² - 5y + 6), It follows that 17 divides xy – 12x + 15y.