How many 5-digit whole numbers with no 0s are divisible by 6?
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2026-04-24 09:47:04.1777024024
How many 5-digit whole numbers with no 0's are divisible by 6?
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Pick the first three digits arbitrarily (from 1-9). That is $9^3$ possible choices.
Suppose the digit sum of those three digits is a multiple of 3. If we pick the fourth digit as $1,4,7$, then the last digit must be 2 or 8. If we pick the fourth digit as $2,5,8$ then the last digit must be 4. If we pick the fourth digit as $3,6,9$ then the last digit must be 6. So 12 choices for the last pair.
Suppose the digit sum of the first three is a multiple of 3 plus 1. If the fourth digit is $1,4,7$, then the last digit must be 4. If the fourth digit is $2,4,8$ then the last digit must be 6. If the fourth digit is $3,6,9$, then the last digit must be $2,8$. Again 12 choices.
Similarly if the digit sum of the first three is a multiple of 3 plus one, we again get 12 choices: fourth digit $1,4,7$ must be followed by 4; $2,5,8$ must be followed by 6; $3,6,9$ must be followed by $2,8$.
So total $12\cdot9^3$.