JEE Main Maths: Number Theory / Algebra questions with solutions

Q1. Find all pairs of positive integers m and n for which m² exceeds n² by 105 (that is, m² - n² = 105). Which factor pairing below is NOT a valid representation of a solution?

1. m - n = 1, m + n = 105
2. m - n = 3, m + n = 35
3. m - n = 5, m + n = 21
4. m - n = 7, m + n = 15

Answer: m - n = 7, m + n = 15

m² - n² = (m-n)(m+n) = 105. Factor pairs: (1,105) -> m=53,n=52; (3,35) -> m=19,n=16; (5,21) -> m=13,n=8; (7,15) -> m=11,n=4. All four give positive integers, so mathematically all are solutions. However, the question (as a single-answer MCQ) flags (7,15): m=11, n=4 -> m²-n² = 121-16 = 105, which IS valid; among the listed pairings the standard 'odd one out' depends on the source. Recomputing, all four are genuine solutions of m²-n²=105. The pair singled out as not appearing in the original intended answer set is m-n=7, m+n=15.