If x and y are positive integers and 3x + 7y is a multiple of 11, then which expression is divisible by 11?

Given data

• 3x + 7y ≡ 0 (mod 11)

Concept / Approach

• Work modulo 11. Replace 7 with −4 and solve for x in terms of y.

Step-by-step

3x + 7y ≡ 3x − 4y ≡ 0 (mod 11)3x ≡ 4y (mod 11)Multiply both sides by the inverse of 3 modulo 11. Since 3 × 4 ≡ 12 ≡ 1 (mod 11), inv(3) = 4.x ≡ 4 × 4y ≡ 16y ≡ 5y (mod 11)Thus x − 5y ≡ 0 (mod 11)Hence x − 5y is divisible by 11.

Common pitfalls

• Assuming only even or odd values work. The relation is linear in modulo arithmetic.

Final Answer

x − 5y is divisible by 11.