Problem 10 · 2012 Math Kangaroo
Medium
Number Theory
place-valuedigit-sum
The age of Quintus is a two-digit power of five and the age of Sekundus is a two-digit power of two. If one adds the digits of their ages the total obtained is an odd number. How big is the product of the digits of their ages?
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Answer: A — 240
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Hint 1 of 2
A two-digit power of 5 is forced; list the two-digit powers of 2.
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Hint 2 of 2
Use the 'sum of all digits is odd' clue to pick the right power of 2, then multiply the four digits.
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Approach: pin the ages from the power and parity clues
- The only two-digit power of 5 is \(25\) (digits 2 and 5, summing to 7).
- Two-digit powers of 2 are \(16, 32, 64\), with digit sums \(7, 5, 10\).
- Adding each to 7 gives \(14, 12, 17\) — only \(64\) makes the total odd (\(17\)).
- So the four digits are \(2,5,6,4\) and their product is \(2\cdot5\cdot6\cdot4 = 240\), choice A.
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