Suppose a, b are integers such that 2 + a and 35 – b are divisible by 11. Prove that a + b is divisible by 11.

Given: (2 + a) and (35 – b) are divisible by 11, such that, a and b are integers.

To Prove: (a + b) is divisible by 11.

Proof: If (2 + a) and (35 – b) are divisible by 11, then their sum and difference will also be divisible by 11.

Let us take difference of these two numbers.

We can write as,

Difference = (2 + a) – (35 – b) = Divisible by 11

⇒ 2 + a – 35 + b = Divisible by 11

⇒ a + b – 33 = Divisible by 11

Note that, here -33 is absolutely divisible by 11.

⇒ a + b must also be divisible by 11, as a + b – 33 is divisible by 11.

Hence, proved.