# Thread: Factorials & Perfect Squares

1. ## Factorials & Perfect Squares

I've been racking my brain out for the last two days on this problem, but I don't even have the slightest clue on how to approach this problem. Could anybody point me in the right direction please?

Q: Find the values of $n \geq 1$ for which $1! + 2! + ... + n!$ is a perfect square. (Hint: Determine the possibilities for the units digit of a perfect square.)

Using the hint:
$0^2 = 0$
$1^2 = 1$
$2^2 = 4$
$3^2 = 9$
$4^2 = 16$
$5^2 = 25$
$6^2 = 36$
$7^2 = 49$
$8^2 = 64$
$9^2 = 81$
$10^2 = 100$

This leads to a pattern: 0 1 4 9 6 5 6 9 4 1 0

Then I'm just lost....

2. The unit digit of $1!$ is 1.

The unit digit of $2!$ is 2.

The unit digit of $3!$ is 6.

The unit digit of $4!$ is 4.

The unit digit of $n!, \ n\geq 5$ is 0.

Now, we have:
$1!=1=1^2$

$1!+2!+3!=1+2+6=9=3^2$

For $n\geq 4$ the unit digit of $1!+2!+3!+\ldots+n!$ is 3 and it can't be a perfect square.

3. Thanks a lot red_dog.

After your hint, I completed the mathematical proof and finished off the problem.