AMC10 2013 A

AMC10 2013 A · Q13

AMC10 2013 A · Q13. It mainly tests Basic counting (rules of product/sum), Remainders & modular arithmetic.

How many three-digit numbers are not divisible by 5, have digits that sum to less than 20, and have the first digit equal to the third digit?

有多少个三位数不被5整除，数字之和小于20，且首位数字等于末位数字？

(A) 52 52

(B) 60 60

(D) 68 68

(E) 70 70

Answer

Correct choice: (B)

正确答案：（B）

Solution

Each such three-digit number must have the form $aba$, where $a$ and $b$ are digits and $a \neq 0$. Such a number will not be divisible by 5 if and only if $a \neq 5$. If $a$ is equal to 1, 2, 3, or 4, then any of the ten choices for $b$ satisfies the requirement. If $a$ is equal to 6, 7, 8, or 9, then there are 8, 6, 4, or 2 choices for $b$, respectively. This results in $4 \cdot 10 + 8 + 6 + 4 + 2 = 60$ numbers.

每个这样的三位数必须是$aba$形式，其中$a$和$b$是数字且$a \neq 0$。这样的数不被5整除当且仅当$a \neq 5$。如果$a$为1,2,3或4，则$b$有10种选择满足要求。如果$a$为6,7,8或9，则$b$分别有8,6,4或2种选择。这总共是$4 \cdot 10 + 8 + 6 + 4 + 2 = 60$个数。

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