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

• A. 579
• B. 289
• C. 580
• D. 290

Answer 1

Answer: (A)

579

Answer 2

Given :
am × bn = 144145 where a > 1 and b > 1.
We can also write 144 as 24 × 32

Therefore, am × bn = 144145 that can be expressed as am × bn = (24 × 32)145
= 2580 × 3290

As we know that 3290 is a natural number, which implies that it can be expressed as a1, 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.