Divisible by 6 is discussed below:
A number is divisible by 6 if it is divisible by 2 and 3 both.
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.
Consider the following numbers which are not divisible by 6, using the rules of divisibility by 6: 70, 135, 184, 286, 297.
(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.
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