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Ones, Twos and Threes

Age 11 to 14 Short
Challenge Level

Each digit of a positive integer is $1$, $2$ or $3$. Each of these digits occurs at least twice in the number.

What is the smallest such integer that is not divisible by $2$ or $3$?

If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.