### Brain Teasers

# Ordered Triple

Find the number of ordered triples (A, B, C) where A, B and C are positive integers, A is a factor of B, A is a factor of C, and A+B+C=30.

### Hint

If A is a factor of both B and C, A must be a factor of the three numbers' sum 30.### Answer

If A=1, then the possibilities are B=1, C=28; ..., B=28, C=1, 28 possibilities;If A=2, then the possibilities are B=2, C=26; ..., B=26, C=2, 13 possibilities, 41 accumulated;

If A=3, then the possibilities are B=3, C=24; ..., B=24, C=3, 8 possibilities, 49 accumulated;

If A=5, then the possibilities are B=5, C=20; ..., B=20, C=5, 4 possibilities, 53 accumulated;

If A=6, then the possibilities are B=6, C=18; ..., B=18, C=6, 3 possibilities, 56 accumulated;

If A=10, then the only possibility is B=10, C=10, 57 possibilities accumulated.

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