Find $x$ !

Number Theory Level 1

$x$ is a positive integer satisfying $\dfrac{10! + 11! + 12!}{10!} = x^{2}$ .

Find the value of $x$ .

Notation :
$!$ denotes the factorial notation. For example, $8! = 1\times2\times3\times\cdots\times8$ .

1 solution

Raihan Fauzan
May 6, 2016

$x^{2} = \frac{10! + 11! + 12!}{10!}$

$x^{2} = \frac{10!(1 + 11 + 11 \times 12)}{10!}$

$x^{2} = 1 + 11 + 132$

$x^{2} = 144$

$x = \sqrt{144}$

$x = \boxed{12}$

This can be generalised for all $x$ that

$\frac { (x - 2)!+(x - 1)!+x! } { x! } = x^2$

Goh Choon Aik - 4 years, 10 months ago