Maths circle: Remainders – We Solve Problems

#### Problem № Maths circle: Remainders

Some time ago we talked about remainders in division by $3$, before that we discussed parity. Now it is time for some more general remainder problems. Remainder is a number that is “left over” from division. Even if a number $n$ is not divisible by another number $k$, it is still possible to divide $n$ by $k$, but with a remainder $r$. Then, we can write $n = qk +r$. It is important that $r$ is never larger or equal to $k$. That is because we say that $k$ goes into $n$ $q$ times, and a little bit is left. If that little bit was larger than $k$, it could “go into” $n$ once more. For example, a remainder of $44$ in division by $7$ is $2$, because $44 = 6 \times 7 + 2$. \par
The remainders can be quite powerful when we learn how to add, subtract and multiply them. We can then use them to learn about properties of numbers. $\\$
The general rule is that a $\textbf{remainder}$ of a sum, difference or a product of two remainders is equal to the $\textbf{remainder}$ of a sum, difference or a product of the original numbers. What that means is if we want to find a remainder of a product of two numbers, we need to look at the individual remainders, multiply them, and then take a remainder. For example, $10$ has a remainder $3$ in division by $7$ and $11$ has a remainder $4$ in division by $7$. The product $10 \times 11 = 110$ will have the same remainder as the product of the individual remainders. We first multiply $3 \times 4 =12$ and then take a remainder in division by $7$, that is $5$, because $12 = 7+5$. That means that $110$ gives a remainder $5$ in division by $7$ – and it does, because $110 = 15 \times 7 + 5$. If a number is divisible by a number we are dividing it, nothing remains and we say the remainder is $0$. $\\$
Let’s have a look on some examples:

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