Maths circle: Remainders – We Solve Problems

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:

Problems:

1.

Division with remainder , Division with remainders. Arithmetic of remainders

What time is it going to be in $2019$ hours from now?

2.

Division with remainder , Division with remainders. Arithmetic of remainders

What is a remainder of $1203 \times 1203 – 1202 \times 1205$ when divided by $12$?

3.

Division with remainder , Division with remainders. Arithmetic of remainders

Show that a perfect square can only have remainders $0$ or $1$ when divided by $4$.

4.

Arithmetic of remainders , Division with remainders. Arithmetic of remainders

Convert $2000$ seconds into minutes and seconds.

5.

Division with remainder , Division with remainders. Arithmetic of remainders

What is a remainder of $7780 \times 7781 \times 7782 \times 7783$ when divided by $7$?

6.

Division with remainder , Division with remainders. Arithmetic of remainders

Tim had more hazelnuts than Tom. If Tim gave Tom as many hazelnuts as Tom already had, Tim and Tom would have the same number of hazelnuts. Instead, Tim gave Tom only a few hazelnuts (no more than five) and divided his remaining hazelnuts equally between $3$ squirrels. How many hazelnuts did Tim give to Tom?

7.

Division with remainder , Division with remainders. Arithmetic of remainders

Prove that $n^3 – n $ is divisible by $24$ for any odd $n$.

8.

Division with remainder , Division with remainders. Arithmetic of remainders

Show that if numbers $a-b$ and $c-d$ are divisible by $11$, then $ac-bd$ and $ad – bc$ are also both divisible by $11$.

9.

Division with remainder , Division with remainders. Arithmetic of remainders

For how many pairs of numbers $x$ and $y$ between $1$ and $100$ is the expression $x^2 + y^2$ divisible by $7$?

10.

Division with remainder , Division with remainders. Arithmetic of remainders

Seven robbers are dividing a bag of coins of various denominations. It turned out that the sum could not be divided equally between them, but if any coin is set aside, the rest could be divided so that every robber would get an equal part. Prove that the bag cannot contain $100$ coins.

11.

Division with remainder , Division with remainders. Arithmetic of remainders

Show that the equation $x^2 +6x-1 = y^2$ has no solutions in integer $x$ and $y$.

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