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No Positive Integers with a^2 - b^2 = 18 | IOQM

JEE Maths question with a full step-by-step solution.

Question
The number of all positive integers aa and bb such that a2−b2=18a^2 - b^2 = 18.
Solution
Answer: 0
Step 1: Factor. The equation a2−b2=18a^2 - b^2 = 18 gives (a−b)(a+b)=18(a - b)(a + b) = 18. Step 2: Parity restriction. Since (a−b)+(a+b)=2a(a - b) + (a + b) = 2a is even, a−ba - b and a+ba + b have the same parity. Step 3: If both are odd, their product is odd, which cannot equal 1818. Step 4: If both are even, their product is a multiple of 44, but 1818 is not a multiple of 44. Step 5: Both possibilities fail, so a2−b2=18a^2 - b^2 = 18 has no solution in positive integers. Answer: There are no positive integer solutions.
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