Let’s talk about congruence and division two concepts that might seem like they don’t belong together, but trust us, they do.

Before anything else: what is congruence? Well, let’s say you have two numbers that are equal when divided by another number (called the modulus). For example, if I tell you that 8 is congruent to 23 modulo 5, it means that both of these numbers leave a remainder of 3 when they are divided by 5. So instead of saying “8 equals 23 when we divide them by 5 and take the remainder,” we can just say “8 is congruent to 23 modulo 5.”

Now, division with congruence. Unfortunately, it’s not as straightforward as regular division because we cannot always divide two numbers using congruences. For example, if I tell you that 10 is congruent to 4 modulo 3, and then ask you what happens when you divide 2 by 10 (which is congruent to 1 modulo 3), it’s not as simple as saying “you get a remainder of 2.” That would be incorrect!

Here’s why: if we try to divide 2 by 10, which is congruent to 1 modulo 3, using regular division, we end up with a quotient and a remainder. Let’s say the quotient is q and the remainder is r. Then we can write this as:

2 = (q * 10) + r

But since 10 is congruent to 1 modulo 3, that means any number multiplied by 10 will also be congruent to it modulo 3. So if we substitute the value of 10 in our equation above with its equivalent (which is 1), we get:

2 = (q * 1) + r

Now, let’s say that q and r are both integers. Then this means that when we divide 2 by 10 using congruence modulo 3, the remainder will always be either 2 or -1. That’s because if we take any integer value for q (let’s call it x), then:

x * 1 + r = 2

But since both x and r are integers, that means their sum must also be an integer. And the only way to get a remainder of 2 when dividing by 3 is if we add either -1 or 2 to our quotient (which is x). So in this case, there’s no unique answer for what happens when you divide 2 by 10 using congruence modulo 3.

But don’t worry! There are still some cases where division with congruences works just fine. For example, let’s say we have the following congruence:

6^41 is congruent to 3 modulo 7

We can use this fact to simplify an expression like (6^2)^20 * 6^5. Let’s break it down:

(6^2)^20 = (6*6)^20

Now, let’s apply the power rule for congruences and see what happens when we take this to the modulo 7:

(6*6) ^ 20 is congruent to (1 * 1) ^ 20 which is just 1^20. So that means our expression simplifies to:

1^20 * 6^5 = 1 * 6^5 modulo 7

And since we know that 3 is congruent to 6 modulo 7, this means that the final answer will be either 3 or 4 (depending on whether we add a multiple of 7 to our quotient). Congruence and division can work together in some cases, but not all. Just remember to always check your answers using regular division before relying solely on congruences for your calculations.