Dear Uncle Colin,

My six siblings and I have inherited a fortune of £$10^{10} + 10^{(10^2)} + \dots 10^{(10^{10})}$, to be divided evenly between us. However, we’re a very squabbly family, so we want to know how much money will be left over once it’s divided up. Can you help?

- Remaining Inheriting Causes Headaches

Hi, RICH, and thanks for your message. I’m, er, sorry for your loss.

Obviously, you could add up all of the money amounts, divide by seven and see what’s left over - but that would take forever. So let’s do it more logically.

Breaking it down

We all know that 1001 is a multiple of 7, so $10^3 \equiv -1 \pmod{7}$. As a result, $10^9 \equiv -1 \pmod{7}$, which means not only that $10^{10} \equiv 4 \pmod{7}$, but so does $10^{(10^k)}$ for any positive integer $k$.

That means, each of the ten terms of your series gives a remainder of 4 when divided by seven.

Overall, that’s 40 spare pounds at the end. Putting those in a pile and splitting them between the seven of you uses up all but five of them - so your remainder is five.

I hope that helps!

- Uncle Colin