Are you ready for some serious number crunching? Let’s talk about congruence and arithmetic the coolest way to play with numbers without actually doing any real work (just kidding, we still have to do some calculations).

So what is this magical concept of “congruence” you ask? Well, it’s basically a fancy word for saying that two numbers are equivalent in terms of their remainder when divided by another number. Let me explain with an example: let’s say you want to buy a pack of gum at the store but only have $5 and the gum costs $3. You can either pay with your $5 bill or use 2 one-dollar bills instead (which is equivalent in terms of value). In math speak, we would write this as:

$5 \equiv 2(mod1)$

This means that both $5$ and $2+1+1$ have a remainder of $0$ when divided by $1$. So they are congruent modulo $1$, which is just fancy math talk for saying they’re the same in terms of their “leftover” part.

Now let’s say you want to buy that gum but only have a $20 bill and some change. The cashier tells you that the gum costs exactly $3, so you can either pay with your $20 or use two one-dollar bills instead (which is equivalent in terms of value). In math speak, we would write this as:

$20 \equiv 2(mod1)$

This means that both $20$ and $2+1+1$ have a remainder of $0$ when divided by $1$. So they are congruent modulo $1$, which is just fancy math talk for saying they’re the same in terms of their “leftover” part.

But what if you want to buy that gum but only have a $25 bill and some change? Well, unfortunately, there’s no way to make exact change with two one-dollar bills (unless you get lucky and the cashier gives you back a penny). In math speak, we would write this as:

$25 \not\equiv 3(mod1)$

This means that $25$ is not congruent to $3$ modulo $1$. So they are not equivalent in terms of their “leftover” part.

Now some cool properties of congruence. For example, if a number is congruent to another number modulo n, and we add or subtract the same amount from both numbers, then the resulting numbers will also be congruent modulo n! Let me show you an example:

$5 \equiv 1(mod2)$

This means that $5$ and $1$ have a remainder of $1$ when divided by $2$. Now let’s add $3$ to both numbers:

$8 \equiv 4(mod2)$

This means that $8$ and $4$ have a remainder of $0$ when divided by $2$. So they are congruent modulo $2$, which is just fancy math talk for saying they’re the same in terms of their “leftover” part.

In other words, if we add or subtract the same amount from both numbers, then the resulting numbers will have the same remainder when divided by n (which means they are congruent modulo n). This is a really cool property that can help us simplify calculations and make our lives easier!