Tests of Divisibility

1. Divisibility By 2 :

A number is divisible by 2, if its unit’s digit is any of 0, 2, 4, 6, 8.

Example:

84932 is divisible by 2,  while 65935 is not.

2. Divisibility By 3 :

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

Example:

592482 is divisible by 3, since sum of its digits = (5 + 9 + 2 + 4 + 8 + 2)=30, which is divisible by 3.

But, 864329 is not divisible by 3, since sum of its digits =(8 + 6 + 4 + 3 + 2 + 9) =32, which is not divisible by 3.

3. Divisibility By 4 :

A number is divisible by 4, if the number formed by the last two digits is divisible by 4.

Example:

892648 is divisible by 4, since the number formed by the last two digits is 48, which is divisible by 4.

But, 749282 is not divisible by 4, since the number formed by the last two digits is 82, which is not divisible by 4.

4. Divisibility By 5 :

A number is divisible by 5, if its unit’s digit is either 0 or 5.

Example:

Thus, 20820 and 50345 are divisible by 5, while 30934 and 40946 are not.

5. Divisibility By 6 :

A number is divisible by 6, if it is divisible by both 2 and 3.

Example:

The number 35256 is clearly divisible by 2. Sum of its digits = (3 + 5 + 2 + 5 + 6) = 21, which is divisible by 3. Thus, 35256 is divisible by 2 as well as 3.
Hence, 35256 is divisible by 6.

6. Divisibility By 8 :

A number is divisible by 8, if the number formed by the last three digits of the given number is divisible by 8.

Example:

953360 is divisible by 8, since the number formed by last three digits is 360, which is divisible by 8.

But, 529418 is not divisible by 8, since the number formed by last three digits is 418, which is not divisible by 8.

7. Divisibility By 9 :

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

Example:

60732 is divisible by 9, since sum of digits, (6 + 0 + 7 + 3 + 2) = 18, which is divisible by 9.

But, 68956 is not divisible by 9, since sum of digits = (6 + 8 + 9 + 5 + 6) = 34, which is not divisible by 9.

8. Divisibility By 10 :

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

Example:

96410, 10480 are divisible by 10, while 96375 is not.

9. Divisibility By 11 :

A number is divisible by 11, if the difference of the sum of its digits at odd places and the sum of its digits at even places, is either 0 or a number divisible by 11.

Example:

The number 4832718 is divisible by 11, since : (sum of digits at odd places) – (sum of digits at even places) = (8 + 7 + 3 + 4) – (1 + 2 + 8) = 11, which is divisible by 11.

10. Divisibility By 12

A number is divisible by 12, if it is divisible by both 4 and 3.

Example:

Consider the number 34632.
(i) The number formed by last two digits is 32, which is divisible by 4,
(ii) Sum of digits = (3 + 4 + 6 + 3 + 2) = 18, which is divisible by 3. Thus, 34632 is divisible by 4 as well as 3.
Hence, 34632 is divisible by 12.

11. Divisibility By 14 :

A number is divisible by 14, if it is divisible by both 2 and 7.

12. Divisibility By 15 :

A number is divisible by 15, if it is divisible by both 3 and 5.

13. Divisibility By 16 :

A number is divisible by 16, if the number formed by the last 4 digits is divisible by 16.

Example:

7957536 is divisible by 16, since the number formed by the last four digits is 7536, which is divisible by 16.

14. Divisibility By 24 :

A given number is divisible by 24, if it is divisible by both3 and 8.

15. Divisibility By 40 :

A given number is divisible by 40, if it is divisible by both 5 and 8.

16. Divisibility By 80 :

A given number is divisible by 80, if it is divisible by both 5 and 16.

Notes:

1. When a number is divisible by another number, it is also divisible by the factors of the number.

For example, 48 is divisible by 12

And 12 = 2 × 3 × 4, so 2, 3 and 4 are the factors of 12.

Therefore 48 is also divisible by 2, 3 and 4

i.e. 48 ÷ 2 = 24, 48 ÷ 3 = 16 and 48 ÷ 4 = 12

2. When a number is divisible by two or more co-prime numbers, it is also divisible by their products.

For example, 12 is divisible by 2 and 3 as, 12 ÷ 2 = 6 and 12 ÷ 3 = 4

The product of 2 and 3 = 2 × 3 = 6

Therefore, 12 is also divisible by 6 i.e., 12 ÷ 6 = 2.

3. When a number is a factor of two given numbers, it is also a factor of their sum and difference.

For example, 6 is a factor of 18 since 18 ÷ 6 = 3

6 is a factor of 30 since 30 ÷ 6 = 5

Also, 6 is a factor of (30 + 18) = 48 since, 48 ÷ 6 = 8.

Also, 6 is a factor of (30 – 18) = 12 since, 12 ÷ 6 =2.

4. When a number is a factor of another number, it is also a factor of any multiple of that number.

For example, 4 is a factor of 12 since, 12 = 4 × 3.

Also, 108 is multiple of 12 as 108 = 9 × 12.

Therefore, 12 is also factor of 108 since 108 = 4 × 27.

Sample Questions

Q1. Is 4832718 divisible by 11? ( Question can also come in the format: Which one out of these is divisible by 11)

Sol: (Sum of digits at odd places) – (Sum of digits at even places)= (8 + 7 + 3 + 4) – (1 + 2 + 8) = 11,
which is divisible by 11.

Hence, 4832718 is divisible by 11.

Q2. 476 ** 0 is divisible by both 3 and 11. The non-zero digits in the hundred’s and ten’s places are respectively:
A. 7 and 4
B. 7 and 5
C. 8 and 5
D. None of these

Answer:
Let the given number be 476 xy 0.
Then (4 + 7 + 6 + x + y + 0) = (17 + x + y) must be divisible by 3.
And, (0 + x + 7) – (y + 6 + 4) = (x – y -3) must be either 0 or 11.
x – y – 3 = 0 y = x – 3
(17 + x + y) = (17 + x + x – 3) = (2x + 14)
x= 2 or x = 8.
x = 8 and y = 5.

Q3. When a number is divided by 13, the remainder is 11. When the same number is divided by 17, then remainder is 9. What is the number ?
A. 339
B. 349
C. 369
D. Data inadequate

Answer
x = 13p + 11 and x = 17q + 9
13p + 11 = 17q + 9
17q – 13p = 2
q = (2 + 13p)/17
The least value of p for which q = (2 + 13p)/17 is a whole number is p = 26
x = (13 x 26 + 11) = (338 + 11)
= 349

Q4. On multiplying a number by 7, the product is a number each of whose digits is 3. The smallest such number is:
A. 47619
B. 47719
C. 48619
D. 47649

Answer
By hit and trial, we find that
47619 x 7 = 333333.

Q5. A number was divided successively in order by 4, 5 and 6. The remainders were respectively 2, 3 and 4. The number is:
A. 214
B. 476
C. 954
D. 1908

Answer

[4 | x] z = 6 x 1 + 4 = 10
[5 | y -2] y = 5 x z + 3 = 5 x 10 + 3 = 53
[6 | z – 3] x = 4 x y + 2 = 4 x 53 + 2 = 214
| 1 – 4
Hence, required number = 214

Q6. The least number that must be subtracted from 999,999 so that it becomes divisible by 125:
a) 4
b) 124
c) 24
d) None of these

Answer

999,999 is very close to 1000,000 and the later is divisible by 125. [ To check the divisibility of 125 we need to check the divisibility of last 3 digits]. We can easily verify that the next multiple of 125 less than 1000,000 will be (1000,000 -125). Since, the actual number is 999,999, the number that needs to be subtracted is 124

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