AMC 8 2000 Problem 11
AMC8 2000 年第 11 题
整除与因数★★★☆☆
The number 64 has the property that it is divisible by its unit digit. How many whole numbers between 10 and 50 have this property?
64 这个数有这样的性质:它可以被它的个位所整除，那么 10 和 50 之间有多少个整数有这样的性质？
- A.15
- B.16
- C.17
- D.18
- E.20
正确答案 · Correct Answer
C
解析 · Solution
Solution 1
Casework by the units digit will help organize the answer. u=0 gives no solutions, since no real numbers are divisible by 0. u=1 has 4 solutions, since all numbers are divisible by 1. u=2 has 4 solutions, since every number ending in 2 is even (ie divisible by 2 ). u=3 has 1 solution: 33. or will retain the units digit, but will stop the number from being divisible by 3. is the smallest multiple of 10 that will keep the number divisible by 3, but those numbers are 3 and 63, which are out of the range of the problem. u=4 has 2 solutions: 24 and 44. Adding or subtracting 10 will kill divisibility by 4, since 10 is not divisible by 4. u=5 has 4 solutions: every number ending in 5 is divisible by 5. u=6 has 1 solution: 36. or will kill divisibility by 3, and thus kill divisibility by 6. u=7 has no solutions. The first multiples of 7 that end in 7 are 7 and 77, but both are outside of the range of this problem. u=8 has 1 solution: 48. will all kill divisibility by 8 since 10, 20, and 30 are not divisible by 8. u=9 has no solutions. 9 and 99 are the smallest multiples of 9 that end in 9. Totalling the solutions, we have 0 + 4 + 4 + 1 + 2 + 4 + 1 + 0 + 1 + 0 = 17 solutions, giving the answer , which is 17.
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