Digital division
How many three digit numbers formed with three different digits from 0, 1, 2, 3 and 5 are divisible by 6?
Age
11 to 14
Challenge level
Problem
Consider all three-digit numbers formed by using different digits from 0, 1, 2, 3 and 5. How many of these numbers are divisible by 6?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
A number is a multiple of 6 precisely when it is both a multiple of 2 and of 3. To be a multiple of 2, it will need to end with an even digit; i.e. 0 or 2. To be a multiple of 3, the sum of the digits has to be a multiple of 3.
If it ends with 0, the sum of the other two digits must be a multiple of 3; and only $3 = 1 + 2$ or $6 = 1 + 5$ are possible. That gives the numbers 120, 210, 150, 510.
If it ends with 2, the sum of the others must be $1 = 0 + 1$ or $4 = 1 + 3$. That gives 102, 132 and 312.
Hence 7 of these numbers are divisible by 6.