Dear Uncle Colin,

I need to find four consecutive numbers such that the first is a multiple of 5, the second a multiple of 7, the third a multiple of 9 and the fourth a multiple of 11. Can you find such a number?

- Summing Up Needs Zero Intelligence

Hi, SUNZI, and thanks for your message! There are several ways to do this, including “write a computer script”. Instead, I’m going to use a theorem, and then show you my very clever friend Alison’s method.

The Chinese Remainder Theorem

I think it should be called Sunzi ((hey! That’s your name!))’s Remainder Theorem, personally.

Whatever we call it, the upshot is that we can build our number up one piece at a time. Here’s how.

The first condition tells us we have a number that’s a multiple of 5 that is one less than a multiple of 7. A quick flick through the times tables gives 20 as an answer; any number that’s 20 more than a multiple of 35 will satisfy the first two conditions. (In fact, it’s a necessary condition, but I won’t prove that here.)

To find a number that’s congruent to 20, modulo 35 and to 7, modulo 9, we can think about adding on multiples of 35 to 20 until we get a number that works.

20 is congruent to 2, modulo 9; 35 is congruent to 8 (or rather, -1). If we add on four 35s, we’ll get something that’s congruent to 20, modulo 35 and to 7 (or -2) modulo 9. The number we get is 160; our answer needs to be congruent to 160, modulo 315.

Lastly, we need it to be congruent to 8, modulo 11. 160 is congruent to 6; 315 is congruent to 7. It turns out that $6 + 5 \times 7 \equiv 8 \pmod {11}$, so $160 + 5\times 315 = 1735$ satisfies all of the conditions. You can check it if you want to!

(Adding any multiple of 3465 will also work.)

Alison’s way

  • $3465 + 5$ is an even multiple of 5
  • $3465 + 7$ is an even multiple of 7
  • $3465 + 9$ is an even multiple of 9
  • $3645 + 11$ is an even multiple of 11

These four numbers are separated by a common difference of 2 - so $\frac{3470}{2} = 1735$ is a number that’s a multiple of 5 followed by a multiple of 7, then 9, then 11.

How nice is that?

I hope that helps!

- Uncle Colin