Problem 9 · 2013 Math Kangaroo
Medium
Number Theory
divisibility
The number 36 has a special property: 36 can be divided by its units digit with no remainder (36 is divisible by 6). The number 38 does not have this property. How many numbers between 20 and 30 have the same property as 36?
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Answer: C — 4
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Hint 1 of 3
Check each number from 21 up to 29 one at a time.
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Hint 2 of 3
For each one, share it into groups the size of its last digit and see if there is any leftover.
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Hint 3 of 3
Count only the numbers that split evenly with nothing left over.
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Approach: check divisibility by the units digit
- Go through 21 to 29 and divide each by its last digit: \(21\div1\), \(22\div2\), \(24\div4\) and \(25\div5\) all come out even with no leftover.
- The rest (23, 26, 27, 28, 29) each leave a leftover.
- So 4 numbers have the property, which is choice C.
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