Question

How many natural numbers $n$ are there such that $n!+10$ is a perfect square?

• A. $1$
• B. $2$
• C. $4$
• D. Infinitely many.

Answer 1

Answer: (A)

$1$

If $n=1,2,4,5$ then $n!+10$ is not prefect square

If $n=3,$ $n!+10=16$ which is a perfect square

For $n>5,$

$n!+10=(1\times 2\times 3\times 4\times 5\times 6\times ....\times n)+10$

Taking $10$ common

$n!+10=(1\times 2\times 3\times 4\times 5\times 6\times ....\times n)+10=10\left[\right(1\times 3\times 4\times \mathrm{6....}\times n)+1]=2\times 5\left[\right(1\times 3\times 4\times \mathrm{6....}\times n)+1]$

You can clealry see term outside the brackets is even and inside is odd, so the exponent of $2$ will always remain $1$

So, $n!+10$ can not be a perfect square for $n>5$

So, option A is correct as we got one value of $n$ which is $3$.