The highest common factor (H.C.F) of 3240, 3600, and an unknown third number is 36, while their least common multiple (L.C.M) equals 24 × 35 × 52 × 72. What is the third number?

Explanation

Given 3240 = 2³ × 3⁴ × 5, and 3600 = 2⁴ × 3² × 5², their H.C.F is 36 = 2² × 3². Since the H.C.F is the product of the lowest powers of common prime factors, the third number must contain at least 2² × 3². The L.C.M is the product of the highest powers of all prime factors, which includes 2⁴, 3⁵, 5², and 7². To achieve this L.C.M, the third number must include factors 5 and 7² in appropriate powers. Therefore, the third number is 2² × 3 × 5 × 7².