Divisibility and Remainder-RRB

Divisibility and Remainder-RRB Divisibility and Remainder-RRB

Divisibility and Remainder

The operations addition and multiplication are always possible, but operations subtraction and division without remainder are possible at certain conditions. In subtraction we can easily see if it is possible or not: it is enough the minuend to be higher or equal to the subtrahend. In division it is not always so simple to see if a number is divisible without remainder. There are certain cases of division in which by some indications of the numbers, we can see in advance if the division can be done without remainder or not.

Divisor and dividend

If a number can be divided by another without remainder, we say that the first number is dividend or it is divisible on the second and the second is divisor of the first number.

For example: Number 8 is dividend to 4 and 4 is divisor to 8.But 8 is not dividend to 3 and 3 is not a divisor to 8

15 is dividend to 1, 3, 5, 15 and every one of these numbers is divisor to 15

Every number is dividend to itself. 1 is dividend to all numbers. For example: 4/4 = 1; 7/7 = 1; 9/1 = 1; 12/1 = 12

Divisibility by sum with number

Numbers 6 and 14 are divisible by 2; Their sum 20 is also divisible by 2
Numbers 12, 18, 30 are divisible by 6; Their sum 60 is also divisible by 6

If the addends are divisible individually by a number , their sum is divisible by that number too.

We can use this property of the sum to see if a number is divisible by another, without doing the division

For example: Is 742 divisible by 7?
To answer the question, we expand 742 to 2 addends 742 = 700 + 42. Because both 700 and 42 are divisible by 7 and their sum 742 is also divisible

15 is divisible by 3, but 10 is not divisible by 3. Their sum 25 is not divisible by 3

10 and 30 are divisible by 5, but 8 is not divisible by 5 and their sum 48 is also not divisible by 5

If only one addend is not divisible by a number , the sum is also not divisible by that number

Divisibility by difference with number

40 and 12 are divisible by 4. Their difference 28 is also divisible by 4

If the minuend and the subtrahend are divisible individually by a number , their difference is divisible by that number too.

We use this property to determine if a number is divisible by another one, without making the division
For example: Is 792 divisible by 8? If we add 8 to 792 we get 800, or 800 – 8 = 792

The minuend(800) and the subtrahend(8) are divisible by 8, that is why the difference 792 is divisible by 8 too

40 is divisible by 10, but 12 is not divisible by 10. Their difference 28 is not divisible by 10

40 is not divisible by 6, but 12 is divisible by 6. Their difference 28 is not divisible by 6

If only the minuend or the subtrahend is not divisible by a number, the difference is also not divisible by that number.

Divisibility Rules

We say that a number is divisible if it can be divided evenly with no reminder.

A number is divisible by : 2

If the last digit is even – 0, 2, 4, 6,8.

Example

258 is divisible by 2 because the last digit is 8.

170 is divisible by 2 because the last digit is 0.

If the sum of the digits is divisible by 3.

Example

246 is divisible by 3 because 2 + 4 + 6 = 12 – divisible by 3 (12 = 3 × 4).

954 is divisible by 3 because 9 + 5 + 4 = 18 – divisible by 3 (18 = 3 × 6).

If the last two digits form a number that is divisible by 4.

Example

316 is divisible by 4 because 16 is divisible by 4 (16 = 4 × 4).

528 is divisible by 4 because 28 is divisible by 4 (28 = 4 × 7).

If the last digit is 5 or 0.

Example

135 is divisible by 5 because the last digit is 5.

770 is divisible by 5 because the last digit is 0.

If the number is divisible by both 2 and 3.

Example

282 is divisible by 6 because it is divisible by 2(the last digit is even) and divisible by 3 (2+8+2 = 3 × 4).

780 is divisible by 6 because it is divisible by 2(the last digit is even) and divisible by 3 (7+8+0 = 3 × 5).

If you can double the last digit and subtract the sum from the rest of the number, and get an answer that is divisible by 7(including 0).

Example

203 is divisible by 7 because 20 – 2 ⋅ 3 = 14 – divisible by 7 (2 × 7 = 14).

455 is divisible by 7 because 45 – 2 ⋅ 5 = 35 – divisible by 7 (5 × 7 = 35).

If the last three digits form a number that is divisible by 8.

Example

1888 is divisible by 8 because 888 = 8 × 111.

1112 is divisible by 8 because 112 = 8 × 14.

If the sum of all digits is divisible by 9.

Example

144 is divisible by 9 because 1 + 4 + 4 = 9 and 9 is divisible by 9.

819 is divisible by 9 because 8 + 1 + 9 = 18 and 18 is divisible by 9.

If the number ends in 0.

Example

990 is divisible by 10 because it ends in 0.

2340 is divisible by 10 because it ends in 0.

Divisibility by 2

Integers exactly divisible by 2 are called even; integers not divisible by 2 are called odd.

In the set of natural numbers, for example, numbers 2, 4, 6, 8,…….1000 are even and
numbers 1, 3, 5, 7, 9,……1001 are odd.

0 is considered an even number.

10 is exactly divisible by 2. Every number ending with zero can be represented as a sum of tens.
For example: 30 = 10 + 10 + 10; 50 = 10 + 10 + 10 + 10 + 10

Therefore, because each 10 in the sum is exactly divisible by 2, we can make a conclusion that

every number ending in 0 is divisible by 2.

For example: Numbers 90, 150, 700 are divisible by 2, because they end in 0.

A multi-digit number that does not end in 0 can be presented as a sum of a number ending in 0 and a one-digit number. For example, 596 = 590 + 6

The first addend, 590, is divisible by 2, because it ends in 0. The second addend, 6, is also divisible by 2, hense the number 596 is divisible by 2.

Let us consider 597. We represent it as 590 + 7. Again, the first addend is divisible by 2, but the second one is 7 and it is not divisible by 2. If exactly one of the addends is not divisible by 2, then the sum is not divisible by 2, therefore 597 is not divisible by 2.

So, 596 is divisible by 2 because its last digit is an even number, and 597 is not divisible by 2 because its last digit is an odd number.

Therefore, an integer is even if its last digit represents an even number.

Divisibility by 5

The number 10 is divisible by 5. Every number with 0 as a last digit is also divisible by 5, because we can represent it as a sum of tens.

For example: 40 is divisible by 5, because
40 = 10 + 10 + 10 + 10

As for a number with a different last digit, we can always represent it as a sum of a number with 0 as a last digit and the number of ones.

For example: 425 = 420 + 5
428 = 420 +8

The number 425 is divisible by 5 because the two addends (420 and 5) are both divisible by 5
The number 428 is not divisible by 5 because one of the addends (8) is not divisible by 5.

So, divisibility by 5 depends only on the last digit. If this digit is either 5 or 0, the number is divisible by 5.

A number is divisible by 5 if the number’s last digit is either 0 or 5.

Divisibility by 5 – examples:
The numbers 105, 275, 315, 420, 945, 760 can be divided by 5 evenly.
The numbers 151, 246, 879, 1404 are not evenly divisible by 5.

A quick way to divide numbers by 5

Example 1:

Suppose you need to divide 342 by 5
342 : 5 = 68.4

First step:
Multiply the number by 2.
342 x 2 = 684

Second step:
Move the decimal point one place to the left and you get 68.4

Example 2:

Divide 415 : 5 = ?
First step: 415 x 2 = 830
Second step: 83.0, and your answer is 83.

A quick way to multiply by 5

Let’s multiply quickly without a calculator 62 x 5 = ?

First step:
Divide 62 by 2 – 62:2 = 31

Second step:
Multiply the result by 10.
31 x 10 = 310

So 62 x 5 = 310

Square two-digit numbers ending in 5

Example: 35² = ?

First step:
Multiply the first digit by the next integer: 3 x 4 = 12

Second step:
Append 25 at the end
and your answer is 1225.

Divisibility by 4

The number 100 is divisible by 4. Hence, a number ending with two zeros is divisible by 4.

For example, 500, 700, 300 are divisible by 4 because they end with two zeros. The numbers 6000 and 23 000 are also divisible by 4.

Let us take a number that does not end with zeros, for example, 916. This number can be represented like the sum 900 + 16.
Because both the addends are divisible by 4, their sum 916 is also divisible by 4.

On the contrary, 918 is not divisible by 4 because 918 = 900 + 18 and one of the addends (18) is not divisible by 4.

From the examples we can see that we need only the second addend to determine divisibility by 4. The second addend consists of the last two digits.

So, the number 13724 is divisible by 4 because its last two digits, 24, are divisible by 4.

The number 13722 is not divisible by 4 because its last two digits, 22, are not divisible by 4.

A number is divisible by 4 if the number’s last two digits are zeroes or divisible by 4.

For example, 450, 2506, 15342, 20018 are not divisible by 4.

540, 1256, 32424, 56300 are divisible by 4

Divisibility by 25

100 is divisible by 25.

Therefore, every number that ends in two zeros is divisible by 25

For example: 300, 600, 700, 800, 5000, 9000

Let us consider numbers that do not end in zeros, for instance 8375 or 8345.
We can always represent such numbers as a sum of two numbers, one of which ends with two zeros. For example, 8375 = 8300 + 75 and 8345 = 8300 + 45

In the first sum both numbers 8300 and 75 are divisible by 25, therefore their sum, 8375, is also divisible by 25. On the contrary, in the second sum one of the numbers (45) is not exactly divisible by 25, hence the sum, 8345, is also not divisible by 25.

A number is divisible by 25 if its last two digits are zeros or make a number divisible by 25.

For example: 800, 1100, 34000, 5275, 12825, 14350 are divisible by 25
355, 8640, 12395 are not divisible by 25

Divisibility by 3 and 9

Divisibility by 9

All numbers that contain only the digit 9 are divisible by 9.
For example: 9, 99, 999, 99999

Lets us take a number, for example, 324
324 can be written as a sum of hundreds, tens and ones:
324 = 300 + 20 + 4 or 324 = 100 + 100 + 100 + 10 + 10 + 4
But 100 = 99 + 1 and 10 = 9 + 1
Then 324 = 99 + 99 + 99 + 3 + 9 + 9 + 2 + 4 = (99 + 99 + 99 + 9 + 9)+ (3 + 2 + 4)
The sum inside the first brackets is divisible by 9 because all the addends are divisible by 9. If the sum in the second brackets (3 + 2 + 4) is also divisible by 9, then the whole sum, 324, is divisible by 9.
Since the sum 3 + 2 + 4 is divisible by 9, we conclude that 324 is also divisible by 9.

However, 3 + 2 + 4 is the sum of the digits in our number, hence the rule:

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

For example, 15948 is divisible by 9 because the sum of its digits (1 + 5 + 9 + 4 + 8) is divisible by 9 and 31409 is not divisible by 9 because the sum of its digits (3 + 1 + 4 + 0 + 9) is not divisible by 9.

Divisibility by 3

9 is divisible by 3 =>

Every number divisible by 9 is divisible by 3.

For example, 7425 is divisible by 9, hence it is divisible by 3.

However, a number divisible by 3 is not necessarily divisible by 9. For example 6, 12, 15, 21, 24, 30 are all divisible by 3 but none of them is divisible by 9.

The rule for divisibility by 3 can be easily obtained following the same logic we used with divisibility by 9.

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

For example:
58302 is divisible by 3 because the sum of its digits (5 + 8 + 3 + 0 + 2) is divisible by 3.
69145 is not divisible by 3 because the sum of its digits (6 + 9 + 1 + 4 + 5) is not divisible by 3.