Last-but-one
What is the last-but-one digit of 99! ?
Age
14 to 16
Challenge level
Problem
If all the integers from $1$ up to $n$ are multiplied together, this is called the factorial of $n$, and is written as $n!$.
For example, $4! = 4 \times 3 \times 2 \times 1 = 24$.
This means that the last-but-one digit of $4!$ is $2$.
What is the last-but-one digit of $99!$?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: $0$
$99! = 99 \times 98 \times 97 \times ... \times 3 \times 2 \times 1$.
This product contains the numbers $2$, $5$ and $10$, which multiply together to give $100$.
This means $99!$ is divisible by $100$.
The last two digits of $99!$ are therefore $00$, so the last-but-one digit is $0$.
There are plenty of other sets of numbers which together give the required factor of $100$. Some examples include $10$ and $20$, $5$ and $20$ and $2$ and $50$.