What is the units digit for the number 123^(456) ?
Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the number of noughts in
10 000! and 100 000! or even 1 000 000!
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.