Problem 11 · 2000 AMC 8
Medium
Number Theory
divisibilitycasework
The number 64 has the property that it is divisible by its units digit. How many whole numbers between 10 and 50 have this property?
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Answer: C — 17.
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Hint 1 of 2
The units digit is what you're dividing BY, so sort the numbers by units digit — each digit becomes its own little test. And some digits are free: ending in 1, 2, or 5 *guarantees* divisibility.
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Hint 2 of 2
Why are 1, 2, 5 free? Any number is divisible by 1; any even number (ends in 2) is divisible by 2; any number ending in 5 is divisible by 5. For the remaining digits, you only have a few numbers each to check by hand.
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Approach: casework on the units digit, banking the 'free' ones first
- Free digits: ending in 1, 2, or 5 always divides (rules for 1, 2, 5). In 10–50 each of those digits gives 4 numbers ⇒ 12 winners so far.
- Now hand-check the rest. Digit 3: only 33 works (33÷3=11). Digit 4: 24 and 44 work. Digit 6: only 36 works. Digit 8: only 48 works. Digits 0 (can't divide by 0), 7, and 9 give nothing.
- Total = 12 + 1 + 2 + 1 + 1 = 17.
- Organizing principle: when a property depends on the last digit, splitting into 10 digit-cases turns one big hunt into ten tiny ones — and spotting which cases are automatically true (1, 2, 5) cuts most of the work.
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