Problem 17 · 2013 Math Kangaroo
Stretch
Number Theory
divisibilitycareful-counting
The number 35 has a special property: it can be divided exactly by its units digit, because \(35 \div 5 = 7\). The number 38 does not have this property. How many numbers bigger than 21 but smaller than 30 have this property?
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Answer: B — 3
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Hint 1 of 3
Bigger than 21 and smaller than 30 means the numbers 22, 23, 24, 25, 26, 27, 28, 29.
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Hint 2 of 3
For each one, can you split it into equal groups of its last digit with none left over?
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Hint 3 of 3
Count how many of the eight numbers pass that test.
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Approach: test divisibility by the units digit
- 22 ÷ 2 = 11 (yes), 24 ÷ 4 = 6 (yes), 25 ÷ 5 = 5 (yes).
- 23, 26, 27, 28, 29 do not divide evenly by their last digit.
- So there are 3 such numbers.
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