Divisible by 6 is discussed below:
Consider the multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, .........
All the multiples end with 2, 4, 6, 8 or 0.
Hence, these are divisible by 2.
Number |
Sum of the Digits |
12 18 24 48 54 |
1 + 2 = 3 1 + 8 = 9 2 + 4 = 6 4 + 8 = 12 5 + 4 = 9 |
Since the sums of the digits of all these multiples are divisible by 3, they are divisible by 3.
Numbers which are divisible by both 2 and 3 are divisible by 6.
1. Consider the following numbers which are divisible by 6, using the test of divisibility by 6: 42, 144, 180, 258, 156.
(i) 42
[We know the rules of divisibility by 2, if the unit’s place of the number is either 0 or multiple of 2].
42 is divisible by 2. Since the unit place is 2 which is divisible by 2.
[A number is divisible by 3, if the sum of its digits is a multiple of 3 or divisibility by 3].
42 is divisible by 3. Since the sum of the digits of 42 = 4 + 2 = 6 which is divisible by 3.
Therefore, 42 is divisible by both 2 and 3.
Hence, 42 is divisible by 6.
(ii) 144
[We know the rules of divisibility by 2, if the unit’s place of the number is either 0 or multiple of 2].
144 is divisible by 2. Since the unit place is 4 which is divisible by 2.
[A number is divisible by 3, if the sum of its digits is a multiple of 3 or divisibility by 3].
144 is divisible by 3. Since the sum of the digits of 144 = 1 + 4 + 4 = 9 which is divisible by 3.
Therefore, 144 is divisible by both 2 and 3.
Hence, 144 is divisible by 6.
(iii) 180
[We know the rules of divisibility by 2, if the unit’s place of the number is either 0 or multiple of 2].
180 is divisible by 2. Since the unit place is 0, divisible by 2.[A number is divisible by 3, if the sum of its digits is a multiple of 3 or divisibility by 3].
180 is divisible by 3. Since the sum of the digits of 180 = 1 + 8 + 0 = 9 which is divisible by 3.
Therefore, 180 is divisible by both 2 and 3.
Hence, 180 is divisible by 6.
(iv) 258
[We know the rules of divisibility by 2, if the unit’s place of the number is either 0 or multiple of 2].
258 is divisible by 2. Since the unit place is 8 which is divisible by 2.
[A number is divisible by 3, if the sum of its digits is a multiple of 3 or divisibility by 3].
258 is divisible by 3. Since the sum of the digits of 258 = 2 + 5 + 8 = 15 which is divisible by 3.
Therefore, 258 is divisible by both 2 and 3.
Hence, 258 is divisible by 6.
(v) 156
[We know the rules of divisibility by 2, if the unit’s place of the number is either 0 or multiple of 2].
156 is divisible by 2. Since the unit place is 6 which is divisible by 2.
[A number is divisible by 3, if the sum of its digits is a multiple of 3 or divisibility by 3].
156 is divisible by 3. Since the sum of the digits of 156 = 1 + 5 + 6 = 12 which is divisible by 3.
Therefore, 156 is divisible by both 2 and 3.
Hence, 156 is divisible by 6.
2. Consider the following numbers which are not divisible by 6, using the rules of divisibility by 6: 70, 135, 184, 286, 297.
Note: A number is divisible by 6 if it is divisible by 2 and 3 both.
(i) 70
[We know the rules of divisibility by 2, if the unit’s place of the number is either 0 or multiple of 2].
70 is divisible by 2. Since the unit place is 0 which is divisible by 2.
[A number is divisible by 3, if the sum of its digits is a multiple of 3 or divisibility by 3].
70 is not divisible by 3. Since the sum of the digits of 70 = 7 + 0 = 7 which is not divisible by 3.
Therefore, 70 is divisible by 2 but not by 3.
Hence, 70 is not divisible by 6.
(ii) 135
[We know the rules of divisibility by 2, if the unit’s place of the number is either 0 or multiple of 2].
135 is not divisible by 2. Since the unit place is 5, which is not divisible by 2.
[A number is divisible by 3, if the sum of its digits is a multiple of 3 or divisibility by 3].
135 is divisible by 3. Since the sum of the digits of 135 = 1 + 3 + 5 = 9 which is divisible by 3.
Therefore, 135 is divisible by both 3 but not by 2.
Hence, 135 is not divisible by 6.
(iii) 184
[We know the rules of divisibility by 2, if the unit’s place of the number is either 0 or multiple of 2].
184 is divisible by 2. Since the unit place is 4 which is divisible by 2.
[A number is divisible by 3, if the sum of its digits is a multiple of 3 or divisibility by 3].
184 is not divisible by 3. Since the sum of the digits of 184 = 1 + 8 + 4= 13 which is not divisible by 3.
Therefore, 184 is divisible by 2 but not by 3.
Hence, 184 is not divisible by 6.
(iv) 286
[We know the rules of divisibility by 2, if the unit’s place of the number is either 0 or multiple of 2].
286 is divisible by 2. Since the unit place is 6 which is divisible by 2.
[A number is divisible by 3, if the sum of its digits is a multiple of 3 or divisibility by 3].
286 is not divisible by 3. Since the sum of the digits of 286 = 2 + 8 + 6 = 16 which is not divisible by 3.
Therefore, 286 is divisible by 2 but not by 3.
Hence, 286 is not divisible by 6.
(v) 297
[We know the rules of divisibility by 2, if the unit’s place of the number is either 0 or multiple of 2].
297 is not divisible by 2. Since the unit place is 7, which is not divisible by 2.
[A number is divisible by 3, if the sum of its digits is a multiple of 3 or divisibility by 3].
297 is divisible by 3. Since the sum of the digits of 297 = 2 + 9 + 7 = 18 which is divisible by 3.
Therefore, 297 is divisible by both 3 but not by 2.
Hence, 297 is not divisible by 6.
3. Is 8622 divisible by 6?
The number ends in 2. Hence it is divisible by 2.
Sum of the digits = 8 + 6 + 2 + 2 = 18, is divisible by 3.
Hence 8622 is divisible by 6.
We can verify that 8622 is divisible by 6 by actual division.
Questions and Answers on Divisible by 6:
1. Which of the following numbers are divisible by 6?
(i) 258
(ii) 96420
(iii) 3448
(iv) 1360
(v) 4008
(vi) 771
(vii) 11190
(viii) 6124
Answer:
2. (ii) 96420
(v) 4008
(vii) 11190
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