AMC8 2016
AMC8 2016 · Q5
AMC8 2016 · Q5. It mainly tests Remainders & modular arithmetic.
The number $N$ is a two-digit number.
-- When $N$ is divided by 9, the remainder is 1.
-- When $N$ is divided by 10, the remainder is 3.
What is the remainder when $N$ is divided by 11?
数$N$是一个两位数。
-- 当$N$除以9时，余数是1。
-- 当$N$除以10时，余数是3。
$N$除以11的余数是多少？
(A)
0
0
(B)
2
2
(C)
4
4
(D)
5
5
(E)
7
7
Answer
Correct choice: (E)
正确答案：（E）
Solution
The two-digit numbers that leave a remainder of 3 when divided by 10 are: 13, 23, 33, 43, 53, 63, 73, 83, 93. The two-digit numbers that leave a remainder of 1 when divided 9 are: 10, 19, 28, 37, 46, 55, 64, 73, 82, 91. Among these two sets, 73 is the only common number. When 73 is divided by 11 the remainder is 7.
被 10 除余 3 的两位数有：13、23、33、43、53、63、73、83、93。被 9 除余 1 的两位数有：10、19、28、37、46、55、64、73、82、91。在这两组数中，唯一的公共数是 73。将 73 除以 11，余数为 7。
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