Alright! Today we’re going to talk about congruence modulo properties but first, let’s take a quick break from math and chat about something more important: pizza. Who here loves pizza? (raises hand) Yeah, me too! But have you ever wondered why some people prefer thin crust while others go for thick and chewy? Well, my friends, it all comes down to congruence modulo properties.

Okay, okay let’s get back to math. Congruence modulo is a fancy way of saying that two numbers have the same remainder when divided by another number (called the modulus). For example, 17 and 39 are both equivalent to 2 when we divide them by 8:

17 % 8 = 5 with a quotient of 2
39 % 8 = 1 with a quotient of 4

So in math-speak, we say that 17 is congruent to 39 modulo 8 (written as 17 39 (mod 8)). This means they have the same remainder when divided by 8.

Now some properties of congruence modulo:

Property #1: Reflexivity a is always equivalent to itself, no matter what number we choose as our modulus. For example, 5 5 (mod 7).

Property #2: Symmetry if a is equivalent to b when using some modulus n, then b must also be equivalent to a with the same modulus. In other words, if 10 34 (mod 9), then we can write it as 34 10 (mod 9) too!

Property #3: Transitivity if a is equivalent to b when using some modulus n, and b is equivalent to c with the same modulus, then a must also be equivalent to c. For example, if 27 6 (mod 10), and 6 16 (mod 10), then we can say that 27 16 (mod 10) too!

So why is this all important? Well, congruence modulo properties are used in a variety of math applications from solving systems of linear equations to finding the period of a repeating decimal. But for now, let’s just enjoy the fact that we can use these properties to simplify our calculations and make them easier to understand!

For example, let’s say we want to find x when 3x is congruent to 12 (mod 7). Using property #3, we know that if a is equivalent to b with some modulus n, then a^p is also equivalent to b^p. So instead of calculating 3x directly, let’s use this fact:

3x = 12 (mod 7)
3^2 x = 144 (mod 7)
(3^2 x) % 7 = 6 (since the remainder when we divide 144 by 7 is 6)
So now, instead of calculating 3x directly, we can use this fact to simplify our calculation:

3x = 6 (mod 7)

Using congruence modulo properties, we were able to simplify a complex equation and make it easier to understand. Who says math has to be boring?