Problem 2 · 2014 Math Kangaroo
Easy
Arithmetic & Operations
sum-constraint
Today is Carmen, Gerda and Sabine's birthday. The sum of their ages is now 44. How big will the sum of their ages be the next time it is a two-digit number with two equal digits?
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Answer: C — 77
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Hint 1 of 2
Every year that passes, the total of three ages goes up by 3.
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Hint 2 of 2
Which numbers with two equal digits can you actually reach by adding multiples of 3 to 44?
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Approach: step the total up by 3 each year and land on a repdigit
- Each birthday all three get one year older, so the sum rises by 3 each year.
- Starting from 44, the reachable totals are 44, 47, 50, 53, … (everything 44 + 3k).
- Check the two-equal-digit numbers: 55 and 66 are not of the form 44 + 3k, but 77 = 44 + 33 is.
- So the next time the sum is a two-digit repdigit it equals 77.
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