What is the units digit for the number 123^(456) ?
How many noughts are at the end of these giant numbers?
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
Consider numbers of the form
$$ u(n) = 1! + 2! + 3! +...+ n!. $$
This table shows that the squares of all integers between $0$
and $9$, and therefore all integers, end in either $1$, $4$, $5$,
$6$ or $9$.
No perfect squares end in $3$. Therefore, $u(n)$ with $n > 4$
can never be a perfect square as it has been shown to always end in
All that remains is to find $u(n)$ for all integers of $4$ or
Only $u(1)$ and $u(3)$ are perfect squares, and so they are the
only sums of factorials to be perfect squares.