Answer:
We know that, a number is divisible by 6, if it is divisible by 2 and 3 both. Also, a number is divisible by 2 if it has any of the digits 0, 2, 4, 6 or 8 in its ones place and number is divisible by 3, if the sum of the digits is a multiple of 3. (a) We have, 297144 (i) Divisibility by 2 \[\because \] Units digit of number = 4, so 297144 is divisible by 2. (ii) Divisibility by 3 Sum of digits of given number = 2 + 9 + 7 + 1 + 4 + 4 = 27 \[\because \] 27 is divisible by 3, so 297144 is also divisible by 3. Now, we see that 297144 is divisible by 2 and 3 both. Hence, it is divisible by 6. (b) We have, 1258 (i) Divisibility by 2 \[\because \] Units digit of number = 8, so 1258 is divisible by 2. (ii) Divisibility by 3 Sum of digits of given number = 1 + 2 + 5 + 8 = 16 16 is not divisible by 3, so 1258 is not divisible by 3. Now, we see that 1258 is divisible by 2 but not divisible by 3. Hence, it is not divisible by 6. (c) We have, 4335 (i) Divisibility by 2 \[\because \]Units digit of number = 5 which is not any of the digits 0, 2, 4, 6 or 8. \[\therefore \] 4335 is not divisible by 2. (ii) Divisibility by 3 Sum of digits of given number = 4 + 3 + 3 + 5 = 15 \[\because \] 15 is divisible by 3, so 4335 is divisible by 3. Now, we see that 4335 is divisible by 3 but not divisible by 2. Hence, it is not divisible by 6. (d) We have, 61233 (i) Divisibility by 2 \[\because \] Units digit of number = 3 which is not any of the digits 0, 2, 4, 6 or 8. \[\therefore \] 61233 is not divisible by 2. Now, we have no need to check the given number is divisible by 3 or not because it is not divisible by one of the factors of 6. Hence, the given number 61233 is not divisible by 6. (e) We have, 901352 (i) Divisibility by 2 \[\because \] Units digit of number = 2, so 901352 is divisible by 2. (ii) Divisibility by 3 Sum of digits of given number = 9 + 0 + 1 + 3 + 5 + 2 = 20 \[\because \] 20 is not divisible by 3, so 901352 is not divisible by 3. Now, we see that 901352 is divisible by 2 but not divisible by 3. Hence, it is not divisible by 6. (f) We have, 438750 (i) Divisibility by 2 \[\because \] Units digit of number = 0, so 438750 is divisible by 2. (ii) Divisibility by 3 Sum of digits of given number = 4 + 3 + 8 + 7 + 5 + 0 = 27 \[\because \] 27 is divisible by 3, so 438750 is divisible by 3. Now, we see that 438750 is divisible by 2 and 3 both. Hence, it is divisible by 6. (g) We have, 1790184 (i) Divisibility by 2 \[\because \]Units digit of number = 4, so 1790184 is divisible by 2. (ii) Divisibility by 3 Sum of digits of given number = 1 + 7 + 9 + 0 + 1 + 8 + 4 = 30 \[\because \] 30 is divisible by 3, so 1790184 is divisible by 3. Now, we see that 1790184 is divisible by 2 and 3 both. Hence, it is divisible by 6. (h) We have, 12583 (i) Divisibility by 2 \[\because \] Units digit of number = 3 which is not any of the digits 0, 2, 4, 6 or 8. \[\therefore \] 12583 is not divisible by 2. Now, we have no need to check the given number is divisible by 3 or not because it is not divisible by one of the factors of 6. Hence, the given number 12583 it not divisible by 6. (i) We have, 639210 (i) Divisibility by 2 \[\because \] Units digit number = 0, so 639210 is divisible by 2. (ii) Divisibility by 3 Sum of digits of given number = 6 + 3 + 9 + 2 + 1 + 0 = 21 \[\because \] 21 is divisible by 3, so 639210 is divisible by 3. Now, we see that 639210 is divisible by 2 and 3 both. Hence, it is divisible by 6. (j) We have, 17852 (i) Divisibility by 2 Units digit of number = 2, so 17852 is divisible by 2. (ii) Divisibility by 3 Sum of digits of given number = 1 + 7 + 8 + 5 + 2 = 23 \[\because \] 23 is not divisible by 3, so 17852 is not divisible by 3. Now, we see that 17852 is divisible by 2 but not divisible by 3. Hence, it is not divisible by 6.
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