   SEARCH HOME Math Central Quandaries & Queries  Question from jeff, a parent: find the greatest ten-digit positive multiple of 12 using each digit once and only once? Hi again Jeff,

An integer is divisible by 12 if it is divisible by 3 and 4. An integer is divisible by 3 if and only if the sum of its digits is divisible by 3. Your 10 digit number has digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. The sum of these digits is divisible by 3 so any ten digit number that uses the 10 digits once and only once is divisible by 3.

An integer is divisible by 4 if and only if the 2 digit number made up of its last 2 digits is divisible by 4. Thus 36534632 is divisible by 4 since 32 is divisible by 4 but 5467654 is not divisible by 4.

What is the greatest 10 digit number that uses each digit once and only once? Is this number divisible by 4? Can you rearrange the last 3 digits to produce a 10 digit number that is divisible by 4?

Penny     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.