Sons and Maths
A maths professor once asked his former student what was the age of his sons. The student replied, saying that their sum was 13 and their product was the same as the number on the door. The professor said that this was not enough information and the student told him that his eldest son plays the piano. The professor said thank you and told him their ages. If their ages are , and , from oldest to youngest, what is ?
The answer is 157.
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1 solution
First we need to find out what are combinations of 3 can sum to give 13. They are ( 1 , 1 , 1 1 ) , ( 1 , 2 , 1 0 ) , ( 1 , 3 , 9 ) , ( 1 , 4 , 8 ) , ( 1 , 5 , 7 ) , ( 1 , 6 , 6 ) , ( 2 , 2 , 9 ) , ( 2 , 3 , 8 ) , ( 2 , 4 , 7 ) , ( 2 , 5 , 6 ) , ( 3 , 3 , 7 ) , ( 3 , 4 , 6 ) , ( 3 , 5 , 5 ) , ( 4 , 4 , 5 ) . If we multiply each set, we find that only 2 sets have the same product. They are ( 1 , 6 , 6 ) and ( 2 , 2 , 9 ) . Since the student has an eldest son, the ages are 9, 2 and 2. ( 9 + 2 ) 2 + 9 × 2 × 2 = 1 5 7