AMC 8 2015 Problem 14
AMC8 2015 年第 14 题
奇偶性★★★☆☆
Which of the following integers cannot be written as the sum of four consecutive odd integers?
下面哪个整数无法写成 4 个连续奇数之和？
- A.
- B.
- C.
- D.
- E.
正确答案 · Correct Answer
D
解析 · Solution
Solution 1
Let our 4 numbers be n, n+2, n+4, n+6, where is odd. Then, our sum is 4n+12. The only answer choice that cannot be written as 4n+12, where is odd, is .
If the four consecutive odd integers are $2n-3,
~ 2n-1,+1$ and $2n+3$; then, the sum is $8n$. All the integers are divisible by 8 except.
If the four consecutive odd integers are $a,
~ a+2,+4$ and $a+6$, the sum is $4a+12$, and $4a+12$ divided by 4 gives $a+3$. This means that $a+3$ must be even. The only integer that does not give an even integer when divided by 4 is 100, so the answer is.
From, we have the sum of the 4 numbers to be equal to 4n + 12. Taking mod 8 gives us for some residue and for some odd integer . Since , we can express it as the equation n = 2a + 1 for some integer . Multiplying 4 to each side of the equation yields 4n = 8a + 4, and taking mod 8 gets us , so b = 0. All the answer choices except choice D is a multiple of 8, and since 100 satisfies all the conditions the answer is.
Since they want CONSECUTIVE odd numbers, it won't be hard to just list the sums out: 16=1+3+5+7 40=7+9+11+13 72=15+17+19+21 200=47+49+51+53 All of the answer choices can be a sum of consecutive odd numbers except 100, so the answer is 100.
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