Problem 24 · 2017 Math Kangaroo
Stretch
Number Theory
place-valuedivisibility
A popular two-digit number is made up of the digits a and b. If the number pair is written down three times one after the other, a six-digit number is obtained. This new number is always divisible by
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Answer: C — 7
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Hint 1 of 2
Writing the two-digit number three times equals it times some fixed number.
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Hint 2 of 2
Factor that fixed multiplier 10101.
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Approach: factor the repetition multiplier
- Writing the two-digit number ab three times gives ababab = ab × 10101.
- Factor 10101 = 3 · 7 · 13 · 37, so 7 always divides the result.
- Hence it is always divisible by 7, choice C.
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