perfect squares $a^a$

Number Theory Level 4

How many integers number, $1\leq a \leq 100$ , such that $a^a$ is a perfect square are there?

The answer is 55.

2 solutions

Paola Ramírez
Jan 8, 2015

If $a=2n \rightarrow\ a^{a}$ $=(2n)^{2n}$ so $\sqrt{(2n)^{2n}}$ $=2n^{\frac{2n}{2}}=(2n)^{n}$ thus $a^{a}$ whit $a$ even is perfect square.

For $a=2n-1$ only $a=k^2$ will be perfect square $\rightarrow$ for $a=1,9,25,49,81$ $a^a$ is perfect square.

Total $={(2,4,6...50)} \cup{(1,9,25,49,81)}=\boxed{55 numbers}$

I like how you used set theory to conclude your answer. :)

Prasun Biswas - 6 years, 3 months ago

Good Solution!!

jaiveer shekhawat - 6 years, 5 months ago

@Paola Ramírez , By the way, I noticed it just now that there's a minor typo in the last line. It should be,

$\textrm{Total}=\bigg\lvert\{2,4,6,\ldots,96,98,100\}\cup \{1,9,25,49,81\}\bigg\rvert=55$

Prasun Biswas - 6 years, 1 month ago

Curtis Clement
Jan 9, 2015

All even integers ${a}$ are square numbers, so there are 50 values so far. If ${a}$ is square then so is $a^{a}$ . There are 5 odd squares in the range. $\therefore$ there are $\boxed{55}$ such numbers

perfect squares a a a^a a a

The answer is 55.

2 solutions

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perfect squares $a^a$