Problem 5 · 1997 AJHSME
Medium
Number Theory
digit-sumlisting
There are many two-digit multiples of 7, but only two of them have a digit sum of 10. The sum of these two multiples of 7 is
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Answer: A — 119.
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Hint 1 of 2
You have two conditions — 'multiple of 7' AND 'digits add to 10.' Pick whichever list is SHORTER to write out, then filter it by the other.
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Hint 2 of 2
When two conditions must both hold, generate the smaller set first and test it against the second condition.
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Approach: list the shorter set, filter by the other condition
- There are about thirteen two-digit multiples of 7, so walk through 14, 21, 28, … and keep only those whose digits sum to 10: that gives 28 (2+8) and 91 (9+1).
- Their sum is 28 + 91 = 119.
Another way — start from the digit-sum list instead:
- Two-digit numbers with digit sum 10 are quick to list: 19, 28, 37, 46, 55, 64, 73, 82, 91.
- Test each for divisibility by 7 — only 28 = 7×4 and 91 = 7×13 survive, so the sum is 28 + 91 = 119.
- Lesson: either list works; choosing the shorter condition to enumerate first saves time.
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