Simple. (Problem $3$ )

Algebra Level pending

$x^x - x = x^{x - 1}$

Find the smallest positive integer of $x$ .

3 solutions

Paramananda Das
Jul 8, 2020

Given $x$ is a positive integer we see $x^{x} - x^{x-1} = x \implies x^{x-1} (x-1) = x \cdot 1$ which is possible only if $x-1 = 1 \implies x = 2 .$

Mahdi Raza
Jul 8, 2020

$\Large{2^2 - 2 = 2^{2-1}}$

Did you see my Bus Numbers comment? @Mahdi Raza

Yajat Shamji - 11 months, 1 week ago

I still didn't understand the question

Mahdi Raza - 11 months, 1 week ago

Do you know the Hardy-Ramanujan taxicab number?

Yajat Shamji - 11 months, 1 week ago

Yajat Shamji
Jul 8, 2020

$1^1 - 1 = 1 - 1 = 0 \neq 1^0 = 1$

$2^2 - 2 = 2 = 2^1 = 2$

Therefore, $x = \fbox{2}$ .