Incredible Power

Number Theory Level 2

Given that there exist $2$ distinct positive integers $a$ and $b$ where $1 \lt a$ and $1 \lt b$ such that

$a^{b} = b^{a}$

find the value of $a + b$ .

The answer is 6.

2 solutions

Vishal S
Dec 16, 2014

Since 2^4=4^2 and this is the only possibility by the given condition(1<a and 1<b)

therefore a+b=2+4=6

Sharky Kesa
Aug 21, 2014

I think the question writer was asking for 2 distinct positive integers. Well, it's pretty well known that $2^4 = 4^2$ . $2 + 4 = 6$ .

@Yuxuan Seah

Extra challenge: I know it won't be that difficult for you but can you prove that $6$ is the only solution?

Yuxuan Seah - 6 years, 9 months ago

Your rewording is still incorrect. The LCM of 3 and 3 is 3 which is greater than 1. :D

Sharky Kesa - 6 years, 9 months ago

@Sharky Kesa I think this should work now. Btw, I never edited this question; this is the first time. @Victor Loh were you the one who added the LCM part of the question? :)

Yuxuan Seah - 6 years, 9 months ago

@Yuxuan Seah – No, I didn't.

Victor Loh - 6 years, 9 months ago

Incredible Power

The answer is 6.

2 solutions

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