Discussion

Question:

Let a, b, m and n be natural numbers such that a > 1 and b > 1. If a^mbⁿ = 144¹⁴⁵, then the largest possible value of n - m is

Explanation:

a^mbⁿ = 144¹⁴⁵

⇒ a^mbⁿ = (12²)¹⁴⁵

⇒ a^mbⁿ = (2⁴ × 3²)¹⁴⁵

⇒ a^mbⁿ = 2⁵⁸⁰ × 3²⁹⁰

To maximise n - m, we need to maximise n and minimise m.
∴ Let 3²⁹⁰ = X

⇒ a^mbⁿ = X × 2⁵⁸⁰

∴ a = X, m = 1, b = 2 and n = 580

⇒ n - m = 580 - 1 = 579

Hence, option (a).

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