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Problem of the Week
Problem C and Solution
Five Magnets

Problem

Harlow has five magnets, each with a different number from \(1\) to \(5.\) They arranged these magnets to create a five digit number \(ABCDE\) such that:

Determine the five-digit number that Harlow created.

Solution

Since \(ABC\) is divisible by \(4\), it follows that \(C\) must be even, so \(C=2\) or \(C=4.\)

Since \(BCD\) is divisible by \(5\), it follows that \(D=0\) or \(D=5.\) However, there is no magnet with a \(0\), so it follows that \(D=5.\)

We also know that \(CDE\) is divisible by \(3.\) We can consider the following two cases.

Thus, the three-digit number \(ABC\) is \(AB4.\) The only magnets not used yet are numbered \(1\) and \(2,\) so this number is \(124\) or \(214.\) Since \(214\) is not divisible by \(4,\) but \(124\) is divisible by \(4,\) it follows that \(A=1\) and \(B=2.\)

Therefore, the five-digit number must be \(12453.\)