When a certain positive integer P is divided by another positive integer, the remainder is $${r_{1}}$$ . When a second positive integer Q is divided by the same integer, the remainder is $${r_{2}}$$ and when (P + Q) is divided by the same divisor, the remainder is $${r_{3}}$$ . Then the divisor may be :

Let P = x + $${r_{1}}$$ and Q = y + $${r_{2}}$$, where each of x and y are divisible by the common divisor.
Then, P + Q = (x + $${r_{1}}$$) + (y + $${r_{2}}$$)
                    = (x + y) + ($${r_{1}}$$ + $${r_{2}}$$)
(P + Q) leaves remainder $${r_{3}}$$ when divided by the common divisor.
⇒ [(x + y) + ($${r_{1}}$$ + $${r_{2}}$$) - $${r_{3}}$$] is divisible by the common divisor.
Since (x + y) is divisible by the common divisor, so divisor = $${r_{1}}$$ + $${r_{2}}$$ - $${r_{3}}$$