It allows us to work with numbers efficiently and elegantly when we have big ones that our calculators can’t handle. For example, let’s say you need to calculate the square of 1000 but your calculator only goes up to 999. No problem! Modular congruence has got your back.

Let me explain how it works in a more casual way. If we have two numbers that are multiples of another number (let’s call this the modulus), then they will be “congruent” when you divide them by the modulus and take the remainder. For example, 15 is congruent to 30 (modulo 12) because both 15 and 30 are multiples of 12, and if we divide each number by 12, their remainders will be the same: 3 for both!

Now some useful properties of modular congruence. If you have two numbers that are congruent (let’s call them a and b), then their negatives (-a) and (-b) will also be congruent, because if they were multiples before, adding or subtracting the same multiple won’t change anything!

Another useful property is that if you have two numbers that are congruent (let’s call them a and b), and another number c that is congruent to d modulo n (c d (mod n)), then the sum or difference of these numbers will also be congruent. This is because adding or subtracting multiples won’t change anything!

Finally, the power rule. If you have a number that is congruent to b modulo n (let’s call it x), and another number c that is also congruent to b modulo n (c b (mod n)), then raising either of these numbers to any power will result in a number that is still congruent. This is because multiplying or dividing by multiples won’t change anything!

Modular congruence is like having your own secret code language for math nerds, and it can help us solve problems more efficiently and elegantly than traditional arithmetic. Give it a try next time you’re faced with a big number, and see how much easier your life becomes!