Question

Prove that if n>4, then the number 1!+2!+3!....n! is never a square​

Answer 1

Consider m = n² , for some integer n .

Now observe the fact that m can have only one of the 1,4,5,6,9,0 as its units digit.

Now observe that n! for n > 4 have 0 as its unit digit.

So, sum 1! + 2! + 3! ...n! , for n≥4 has 3 as its unit digit and due to the above fact, this sum can never be a perfect square.