AMC 8 2018 Problem 21
AMC8 2018 年第 21 题
余数与同余★★★☆☆
How many positive three-digit integers have a remainder of 2 when divided by 6, a remainder of 5 when divided by 9, and a remainder of 7 when divided by 11?
有多少个 3 位正整数，满足：当除以 6 余数为 2，当除以 9 余数为 5，当除以 11 余数为 7？
- A.1
- B.2
- C.3
- D.4
- E.5
正确答案 · Correct Answer
E
解析 · Solution
Solution 1
Looking at the values, we notice that 11-7=4, 9-5=4 and 6-2=4. This means we are looking for a value that is four less than a multiple of 11, 9, and 6. The least common multiple of these numbers is , so the numbers that fulfill this can be written as 198k-4, where is a positive integer. This value is only a three-digit integer when is 1, 2, 3, 4 or 5, which gives 194, 392, 590, 788, and 986 respectively. Thus, we have 5 values, so our answer is 5.
Let us create the equations 6x+2 = 9y+5 = 11z+7, and we know , which gives us , the range of the value of . Because 6x+2=11z+7, then 6x=11z+5=6z+5(z+1), so (z+1) must be a multiple of 6. Because 9y+5=11z+7, then 9y=11z+2=9z+2(z+1), so (z+1) must also be a multiple of 9. Hence, the value of (z+1) must be a common multiple of 6 and 9, which means multiples of 18 ( ). So let's say z+1 = 18p; then , so , or . Thus, the answer is 5.
By the Chinese Remainder Theorem, we have that all solutions are in the form x=198k+194 where Counting the number of values, we get 5.
We can use modular arithmetic. Set up the equations: and These equations can also be written as and Since x+4 is congruent to numbers 6, 9, and 11, then it must also be congruent to their LCM. Thus, since 198 is the LCM of 6, 9, and 11. Since these numbers have to be three digits, they can only be 194, 392, 590, 788, and 986. This gives us the answer of 5.
Let be the three digit positive integer. N = 6a + 2 = 9b + 5 = 11c + 7. Then, we add four to all sides and write N + 4 = 6(a+1) = 9(b+1) = 11(c+1). Now, we know that N + 4 is divisible by 6, 9, and 11. The LCM of 6, 9, and 11 is equal to 198, so N = 198k - 4. From this, we can figure out that can be 5 different three digit numbers -- 194, 392, 590, 788, and 986. , the answer is 5.
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