The Magnificent Operator

Number Theory Level 3

Let me define an operator for two natural numbers $a,b$ as $[[]a,b[]]$

$[[]a,b[]]=\frac{a^b}{b^a}\frac{b^b}{a^a}$

Now, if $a+b\leq15$ , with $b\geq8$ , find the number of ordered pairs $(a,b)$ for which $[[]a,b[]]$ is a perfect square of an integer.

The answer is 14.

1 solution

Rushi Wawge
Nov 5, 2014

a^b/b^a*b^b/a^a=(ab)^b/(ab)^a=(ab)^b-a. so b-a=are even for perfect square.(suppose, b-a=6 then (ab)^3^2 so if power is 2 then it is square) if b=8 then for all value of a+b=14 there are only 1 perfect square.similarly there are 14 pairs of (a,b)

The Magnificent Operator

The answer is 14.

1 solution

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