Problem 21 · 2019 Math Kangaroo
Hard
Number Theory
divisibilitycareful-counting
If one of the digits of a two-digit number is deleted, the result in both cases is a factor of the original number. How many two-digit numbers have this property?
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Answer: C — 14
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Hint 1 of 3
Deleting a digit leaves a single digit; the original must be divisible by each remaining digit.
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Hint 2 of 3
Require the number to be divisible by both its tens digit and its units digit.
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Hint 3 of 3
Every multiple of 11 works, plus a few others — count them all.
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Approach: test divisibility by each single digit
- For a number with tens t and units u, both u and t must divide it (u ≠ 0).
- Through 10–99 the ones that pass are 11, 12, 15, 22, 24, 33, 36, 44, 48, 55, 66, 77, 88, 99.
- That is 14 numbers.
Mark:
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