Let a, b, m and n be natural numbers such that a1 and b1. If a^m × b^n = 144145, then the largest possible value of n−m is:

Question

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

UPSC Quizroom · Accepted Answer

Step 1: Maximize n, minimize m

To maximize n − m, take the smallest possible m = 1.

Assign almost all of 144145 to b^n to make n as large as possible.

Step 2: Example factorization

Let a^m = 3290^1 → m = 1

Let b^n = 2^580 → n = 580

Then: n − m = 580 − 1 = 579

Step 3: Conclusion

The largest possible value of n − m is 579

Answer: 579

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