What is the HCF of \(2^{4}\times3^{4}\) and \(2^{5}\times3^{2}\)?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(2^{4}\times3^{2}\)
The HCF contains each common prime only as many times as it occurs in both numbers, so it takes the lower power of each common prime.
Power of 2: lower of 4 and 5 is 4
Power of 3: lower of 4 and 2 is 2
HCF \(=2^{4}\times3^{2}\)
The HCF is \(2^{4}\times3^{2}\), that is, 144.
Why the other options are incorrect
Option (1) \(2^{5}\times3^{3}\) — Incorrect. \(2^{5}\) and \(3^{3}\) do not divide \(2^{4}\times3^{4}\) and \(2^{5}\times3^{2}\) respectively.
Option (3) \(2^{3}\times3^{2}\) — Incorrect. It is a common factor, but not the highest one, since \(2^{4}\) also divides both.
Option (4) \(2^{4}\times3^{4}\) — Incorrect. \(3^{4}\) does not divide \(2^{5}\times3^{2}\).