### More Mods

What is the units digit for the number 123^(456) ?

### N000ughty Thoughts

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!

### Mod 3

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

# Latin Numbers

##### Stage: 4 Challenge Level:

Thank you for this solution to Andrei Lazanu, School 205, Bucharest, Romania and Jonathan Smith, Gresham's School, Norfolk.

Let $N$ be a six digit number with distinct digits. We have to find the number $N$ given that the numbers $N,\ 2N,\ 3N,\ 4N,\ 5N$ and $6N$, when written underneath each other, form a latin square (that is each row and each column contains all six digits).

Let $N = abcdef = 10^5a + 10^4b +10^3c + 10^2d + 10e + f$ and $S = a + b + c + d + e + f$. Then $$21N = N + 2N + 3N + 4N + 5N + 6N = S \times 111111$$

Therefore $$N = S \times 5291.$$

Now $S \geq 1+2 + 3 + 4 + 5 + 6 = 21$. As $6N$ has only $6$ digits it follows that $a=1$. Hence $N \leq 198765$ and $S \leq 198765/5291 \leq 37$. We now check by computing $N = 5291S$ for $21 \leq S \leq 37$ and also $2N$, and visually check the digits of $N$ and $2N$ to see if they are the same. The solution, written as a latin square, is:

 1 4 2 8 5 7 2 8 5 7 1 4 4 2 8 5 7 1 5 7 1 4 2 8 7 1 4 2 8 5 8 5 7 1 4 2