### Brain Teasers

# Math Reunion

Dear Jim:

Our twenty-fifth reunion went wonderfully last week. The Math Department put on a great spread. I'm sorry you couldn't make it. I ran into Ralph Jones there - remember him? It turns out he married Susie Jacobsen a couple of years after graduation - imagine that! They have three kids, all boys, and their ages make a nice puzzle. No two of them are the same age. Right now, Albert's age is the same as the sum of the digits of the ages of his two brothers. A year ago, Bill was in the same situation. Finally, six years from now it will be Charles' turn to have an age equal to the sum of the digits of his brothers' ages. I trust that's enough information to let you figure out their ages.

Give my regards to Sharon, and I hope I'll see you at our next reunion.

Jerry

Our twenty-fifth reunion went wonderfully last week. The Math Department put on a great spread. I'm sorry you couldn't make it. I ran into Ralph Jones there - remember him? It turns out he married Susie Jacobsen a couple of years after graduation - imagine that! They have three kids, all boys, and their ages make a nice puzzle. No two of them are the same age. Right now, Albert's age is the same as the sum of the digits of the ages of his two brothers. A year ago, Bill was in the same situation. Finally, six years from now it will be Charles' turn to have an age equal to the sum of the digits of his brothers' ages. I trust that's enough information to let you figure out their ages.

Give my regards to Sharon, and I hope I'll see you at our next reunion.

Jerry

### Hint

Any non-negative integer is congruent to the sum of its digits, mod 9. (They have the same remainder when divided by 9.)### Answer

Albert is 11, Bill is 6, and Charles is 5, 11 = 6+5. A year ago, they were 10, 5, and 4, 1+0+4 = 5. In six years, they will be 17, 12, and 11, 1+7+1+2=11.You can solve this one by trial and error, but you can also solve simultaneous congruences mod 9. Let a, b, and c represent the present ages of the three boys. Since any non-negative integer is congruent to the sum of its digits, mod 9, we have three congruences:

a = b + c (mod 9)

b-1 = (a-1) + (c-1) (mod 9)

c+6 = (a+6) + (b+6) (mod 9)

Substituting the value of a from the first congruence into the other two and simplifying, we get:

2c = 1 = 10 (mod 9), so c = 5 (mod 9), and

2b = -6 = 12 (mod 9), so b = 6 (mod 9),

so a = b + c = 2 (mod 9).

Age 2 is too young for Albert to be equal to the sum of digits of his brother's ages, and 20 is too old (it would require that both brothers be 19, contrary to the required congruences), so Albert must be 11, and the rest follows quickly.

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