$a^b \times b^a$

Number Theory Level 3

Let $P_N$ denotes the percentage of all positive integers in the range $[1,N]$ that can be written in the form $a^b \times b^a$ , where $a$ and $b$ are positive integers.

Evaluate $\displaystyle \lim_{N\to\infty} P_N$ to 3 decimal places.

For example: $72 = 9 \times 8 = 3^2 \times 2^3$ .

The answer is 100.000.

4 solutions

Jordan Cahn
Oct 5, 2018

Every number $N$ can be written in the form $N^1\times1^N$ .

Abha Vishwakarma
Oct 5, 2018

At first this seemed like a very hard question but after realising that if either $a$ or $b = 1$ then every number can be represented that way. I liked this question!

Andrea Palma
Oct 12, 2018

At first sight i tought it must be 0% couse there are infinitely many primes and there is unique factorisation theorem but soon after i realized 1 as a or b is also allowed and this consideration skyrocketed the answer completely from 0% to 100%. I find it funny!

Akela Chana
Oct 12, 2018

ELEMENTARY WATSON In given equation just put a = 1 and b = any positive integer of your choice .

Yep you can write any positive integer in this form. So answer is 💯.

a b × b a a^b \times b^a a b × b a

The answer is 100.000.

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$a^b \times b^a$