Divisible by 6

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.