### Expenses

What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?

### Factorial

How many zeros are there at the end of the number which is the product of first hundred positive integers?

### Times Right

Using the digits 1, 2, 3, 4, 5, 6, 7 and 8, mulitply a two two digit numbers are multiplied to give a four digit number, so that the expression is correct. How many different solutions can you find?

# More Mods

##### Age 14 to 16Challenge Level

Ian (Coopers Company and Coborn School) sent us the following solution:

In reply to the question More Mods, I have a solution. The units digit of $123^{456}$ is 1. Here is how I got my answer. I used the same method as in the similar question in the January 1999 Six. There is a distinct pattern for the units digit for the powers of 3: 3, 9, 7, 1, 3, 9, ... . As 1 is the 4th out of 4 in the pattern, and as 456 is divisible by 4, it follows that 1 is the units digit.

Focusing on the units digit is the same as working in arithmetic modulo 10 (clock arithmetic) and this is how Oliver of Madras College solved the problem. \eqalign{ 123 &\equiv& 3 &(\mbox{modulo }10) \\ 123^4 &\equiv& (3^4) = 81 \equiv 1 &(\mbox{modulo }10) \\ 123^{456} &\equiv& (123^4)^{114} \equiv 1^{114} = 1 &(\mbox{modulo }10)} So the units digit of $123^{456}$ is 1.