AMC 8 2018 Problem 7

AMC8 2018 年第 7 题

余数与同余★★☆☆☆

The 5-digit number

\underline{2}\;\underline{0}\;\underline{1}\;\underline{8}\;\underline{U}

is divisible by 9. What is the remainder when this number is divided by 8?

5 位数

\underline{2}\,\underline{0}\,\underline{1}\,\underline{8}\,\underline{U}

能被 9 整除。这个数除以 8 所得余数为多少？

正确答案 · Correct Answer

解析 · Solution

Solution 1

We use the property that the digits of a number must sum to a multiple of 9 if it are divisible by 9. This means 2+0+1+8+U must be divisible by 9. The only possible value for

U

then must be 7. Since we are looking for the remainder when divided by 8, we can ignore the thousands. The remainder when 187 is divided by 8 is 3. If you got stuck on this problem, refer to AOPS Number Theory. You're smart. -InterstellerApex, Nivaar Quick addon: 187-160 ( 8×20 ) =27. 27-24 ( 8×3 )= 3. Therefore, the answer is 3.

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