Let $a, b, m$ and $n$ be natural numbers such that $a 1$ and $b 1$ . If $a^m+b^n = 144^{145}$ , 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  $a^m+b^n = 144^{145}$  , then the largest possible value of  $n − m$  is

cdquestions Admin · Accepted Answer

Given : a m × b n = 144 145 where a > 1 and b > 1. We can also write 144 as 2 4 × 3 2 Therefore, a m × b n = 144 145 that can be expressed as a m × b n = (2 4 × 3 2 ) 145 = 2 580 × 3 290 As we know that 3 290 is a natural number, which implies that it can be expressed as a 1 , where a > 1 Therefore, the least possible value of m is 1. By the same logic, the largest value of n is 580. So, the largest value of (n - m) = (580 - 1) = 579 Therefore, the correct option is (A) : 579.

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Let \(a, b, m\) and \(n\) be natural numbers such that \(a >1\) and \(b >1\) . If \(a^m+b^n = 144^{145}\) , then the largest possible value of \(n − m\) is

The Correct Option is A

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