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Notice that shifting a remainder of 7 up by 1 makes it 7 + 1 = 8, and if a number has a “remainder” of 8 when divided by 8, it’s actually a multiple of 8, so its remainder is really 0 mod 8.

This means that the system of congruences

x ≡ 1 mod 5
x ≡ a mod 8

also has a solution less than 40 for every a: For a = 1, the solution is x = 1; for a = 2, the solution is x = 26, and so on.

This reasoning can be generalized to show that

b mod 5
a mod 8

has a solution less than 40 for every a and b, and generalized further to show that every system of congruences of the form

b mod m
a mod n

has a solution less than m × n, as long as m and n are relatively prime. This is the most basic version of the Chinese remainder theorem.

This theorem, like a lot of number theory tricks, is useful in cryptography, the mathematics of encoding and decoding secret messages. For example, you can use the theorem to keep a number secret until a group of people agree to collaborate to identify it.

Suppose you want a number to stay secret unless your friends Alex and Bo agree that they both want to know it. First assign them a pair of relatively prime numbers — say, 13 for Alex and 17 for Bo — whose product is greater than your secret number. Now divide your secret number by each of their numbers and give each of them their individual remainder. Neither will know your number, but they are guaranteed to be able to figure it out together thanks to the Chinese remainder theorem!

Let’s say you tell Alex the number 11 and Bo the number 15. This means that Alex knows your number x satisfies ≡ 11 mod 13 and Bo knows x satisfies ≡ 15 mod 17. Neither of these congruences alone is enough to specify your secret number, but together they form the system

≡ 11 mod 13
≡ 15 mod 17

and the Chinese remainder theorem guarantees that this system has a unique solution less than 13 × 17 = 221. Working together, Alex and Bo can figure out your number is 219.

You probably won’t need the Chinese remainder theorem to plan your next picnic, but in case you need to distribute access to information among your friends or secretly share troop strength with your generals, make sure this extension of least common multiples is on your list.

Exercises

1. Explain why this system of congruences has no solution:

x ≡ 1 mod 4
x ≡ 0 mod 6

Why doesn’t this violate the Chinese remainder theorem?

2. In solving the hot dog problem

x ≡ 1 mod 5
x ≡ 2 mod 8

we used the fact that (-3) × 5 + 2 × 8 = 1. It’s also true that 13 × 5 + (-8) × 8 = 1. What would have happened if we had used this integer combination of 1 instead?

3. Consider this system of congruences:

x ≡ 1 mod 3
x ≡ 2 mod 4

Find three positive solutions to this system. All of these numbers have the same remainder mod 12. What is it?

4. Use the result of exercise 3 to solve this system of congruences:

x ≡ 1 mod 3
x ≡ 2 mod 4
x ≡ 3 mod 5

Click for Answer 1:

Click for Answer 2:

Click for Answer 3:

Click for Answer 4:

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