Dividing and divisibility, factors and multiplies

Remainder after division

The result of division is not always a nice and round number, e.g. $7÷3$. We know that three fits twice into 7. However, $3×2=6$. There is 1 still remaining. We call 1 in this case the remainder or left over after dividing (7 is dividend and 3 is divisor).

To calculate it you need to find the greatest multiple of the divisor which is still lower than the dividend and subtract it from dividend.

Example:

4 is remainder after division by 5:

We write this also as:

The remainder after division must always be smaller than the divisor. For example, when dividing by 8, the remainder can be from 1 to 7. If the dividend is divisible without a remainder, the remainder is 0.

When the dividend is smaller than the the divisor, the quotient is always 0 and the remainder is the dividend itself. For example, in the division $2 ÷ 5 $, the answer is 0 with remainder of 2, because 5 does not fit into 2 at all. 

Remainder after division

The result of division is not always a nice and round number, e.g. $7÷3$. We know that three fits twice into 7. However, $3×2=6$. There is 1 still remaining. We call 1 in this case the remainder or left over after dividing (7 is dividend and 3 is divisor).

To calculate it you need to find the greatest multiple of the divisor which is still lower than the dividend and subtract it from dividend.

Example:

4 is remainder after division by 5:

We write this also as:

The remainder after division must always be smaller than the divisor. For example, when dividing by 8, the remainder can be from 1 to 7. If the dividend is divisible without a remainder, the remainder is 0.

When the dividend is smaller than the the divisor, the quotient is always 0 and the remainder is the dividend itself. For example, in the division $2 ÷ 5 $, the answer is 0 with remainder of 2, because 5 does not fit into 2 at all. 

Remainder after division

The result of division is not always a nice and round number, e.g. $7÷3$. We know that three fits twice into 7. However, $3×2=6$. There is 1 still remaining. We call 1 in this case the remainder or left over after dividing (7 is dividend and 3 is divisor).

To calculate it you need to find the greatest multiple of the divisor which is still lower than the dividend and subtract it from dividend.

Example:

4 is remainder after division by 5:

We write this also as:

The remainder after division must always be smaller than the divisor. For example, when dividing by 8, the remainder can be from 1 to 7. If the dividend is divisible without a remainder, the remainder is 0.

When the dividend is smaller than the the divisor, the quotient is always 0 and the remainder is the dividend itself. For example, in the division $2 ÷ 5 $, the answer is 0 with remainder of 2, because 5 does not fit into 2 at all. 

Remainder after division

The result of division is not always a nice and round number, e.g. $7÷3$. We know that three fits twice into 7. However, $3×2=6$. There is 1 still remaining. We call 1 in this case the remainder or left over after dividing (7 is dividend and 3 is divisor).

To calculate it you need to find the greatest multiple of the divisor which is still lower than the dividend and subtract it from dividend.

Example:

4 is remainder after division by 5:

We write this also as:

The remainder after division must always be smaller than the divisor. For example, when dividing by 8, the remainder can be from 1 to 7. If the dividend is divisible without a remainder, the remainder is 0.

When the dividend is smaller than the the divisor, the quotient is always 0 and the remainder is the dividend itself. For example, in the division $2 ÷ 5 $, the answer is 0 with remainder of 2, because 5 does not fit into 2 at all. 

Dividing by ten

When we divide a number ending in zero by 10, we can remove one zero from the end of the number.

For example:

Both 50 and 10 end in one zero. We can remove one zero from both numbers:

The same rule works with larger numbers:

We remove one zero from both numbers:

Dividing by whole tens

Whole tens are numbers such as 10, 20, 30, 40, 50 and so on. They always end in one zero.

When both the dividend and the divisor end in zero, we can remove the same number of ending zeros from both numbers. Then we divide the remaining numbers.

For example:

Both numbers end in one zero, so remove one zero from each:

Therefore:

The same method works with other whole tens:

Both numbers end in one zero, so remove one zero from each to get $360 ÷ 6$:

Therefore:

Dividing numbers that end in zero

When the dividend ends in zero, we can sometimes temporarily remove the zero, divide the remaining number, and then put the zero back at the end of the result.

For example:

First, ignore the zero at the end of 270:

The original number 270 had one zero at the end, so we put this zero back after 9:

Therefore:

This method works because:

So:

The same idea works with more zeros.

For example:

Temporarily remove the two ending zeros:

Then put the two zeros back:

Therefore:

Long division by a one-digit number without a remainder

Division of larger numbers is difficult, so we use written forms.

The dividend in our example it is 1752 and the divisor is 8. We begin by determining whether the first digit of the dividend (here 1) is greater than or equal to the divisor (here 8). If so, we can divide these numbers. In our case, the first digit of the dividend is smaller than the divisor, so we must also use the second digit in the dividend. So we count how many times 8 fits into 17. We write 2 in the result and calculate the remainder. We multiply the divisor and the result (2×8=16), write the number 16 under 17 and subtract (17−16=1). This remainder must be less than the divisor.

To the remainder (1) we write another digit (5). Since 15 fits 8 only once, we write 1 in the result. We multiply this result again by the divisor (8) and subtract from 15.

We subtract and write another digit (2) to the remainder 7. We know that 8 fits into 72 exactly nine times, so we write 9 in the result and the remainder is 0. The result of the whole operation is 219. In division, we call the result the quotient.

Take a look at another example: 12008÷8

Sometimes it happens that two zeroes occur in the calculation (in our case the remainder after division was 0 and next digit was 0 as well). In this case we write 0 in the quotient and continue with the next number.

If there occurs dividend smaller than divisor (during calculation in the example below there is 00 and 001), we write 0 in the quotient and continue with the next number.

Long division by a one-digit number without a remainder

Division of larger numbers is difficult, so we use written forms.

The dividend in our example it is 1752 and the divisor is 8. We begin by determining whether the first digit of the dividend (here 1) is greater than or equal to the divisor (here 8). If so, we can divide these numbers. In our case, the first digit of the dividend is smaller than the divisor, so we must also use the second digit in the dividend. So we count how many times 8 fits into 17. We write 2 in the result and calculate the remainder. We multiply the divisor and the result (2×8=16), write the number 16 under 17 and subtract (17−16=1). This remainder must be less than the divisor.

To the remainder (1) we write another digit (5). Since 15 fits 8 only once, we write 1 in the result. We multiply this result again by the divisor (8) and subtract from 15.

We subtract and write another digit (2) to the remainder 7. We know that 8 fits into 72 exactly nine times, so we write 9 in the result and the remainder is 0. The result of the whole operation is 219. In division, we call the result the quotient.

Take a look at another example: 12008÷8

Sometimes it happens that two zeroes occur in the calculation (in our case the remainder after division was 0 and next digit was 0 as well). In this case we write 0 in the quotient and continue with the next number.

If there occurs dividend smaller than divisor (during calculation in the example below there is 00 and 001), we write 0 in the quotient and continue with the next number.

Long division by a two-digit number without a remainder

Dividing by a two-digit number is the same as by one-digit number. At the beginning we have to look at the first two digits (33), since we can't fit a two-digit number into a one-digit number (sometimes it is necessary to look at the first three digits instead).

Long division with a remainder

For written division with a remainder, we use the same method as for division without a remainder. In the example 531÷4, we gradually divide the digits from the dividend by the divisor. If we look at the last operation, we see that the division came out with a remainder of 3. We can write that the result of our example is 132 (and remainder is 3). However, it is possible to calculate the exact quotient (instead of using a remainder) by adding a decimal point and zeroes to the dividend (count further outside the set of integers). We would then write the next digits after the decimal point.

Divisibility rules

A number is divisible by 2 if it ends in 0, 2, 4, 6, or 8. These numbers are called even numbers. Numbers not divisible by 2 are called odd numbers.

A number is divisible by 3 if the sum of its digits is divisible by 3.

A number is divisible by 4 if the number's last two digits are divisible by 4 or the number ends with 00.

A number is divisible by 5 if it ends in 0 or 5.

A number is divisible by 6 if the sum of its digits is divisible by 3 and 2.

A number is divisible by 8 if the number's last three digits are divisible by 8 or the number ends with 000.

A number is divisible by 9 if the sum of its digits is divisible by 9.

A number is divisible by 10 if it ends with 0.

Divisibility rules

A number is divisible by 2 if it ends in 0, 2, 4, 6, or 8. These numbers are called even numbers. Numbers not divisible by 2 are called odd numbers.

A number is divisible by 3 if the sum of its digits is divisible by 3.

A number is divisible by 4 if the number's last two digits are divisible by 4 or the number ends with 00.

A number is divisible by 5 if it ends in 0 or 5.

A number is divisible by 6 if the sum of its digits is divisible by 3 and 2.

A number is divisible by 8 if the number's last three digits are divisible by 8 or the number ends with 000.

A number is divisible by 9 if the sum of its digits is divisible by 9.

A number is divisible by 10 if it ends with 0.

Multiples

The multiples of a number are the values in that number’s times table. E.g. the multiples of 5 are 5, 10, 15, 20, 25 and so on.

There is an infinite amount of multiples of any given number.

Factors

Factors of the number are the numbers you multiply together to get this number. A factor is an integer that will divide exactly into another number. E.g. 8 is a factor of 24 because 8 will divide into 24 exactly 3 times with no remainder.

Factor pairs

Factor pairs are two integers which multiply together to make a particular number. For example, the factor pairs of 20 are 1 and 20; 2 and 10; 4 and 5. This means that the factors of 20 are: 1 , 2 , 4 , 5 , 10 and 20. Every integer has a finite number of factors.