AMC8 2015
AMC8 2015 · Q14
AMC8 2015 · Q14. It mainly tests Linear equations, Arithmetic misc.
Which of the following integers cannot be written as the sum of four consecutive odd integers?
以下哪个整数不能表示为四个连续奇整数的和？
(A)
16
16
(B)
40
40
(C)
72
72
(D)
100
100
(E)
200
200
Answer
Correct choice: (D)
正确答案：（D）
Solution
Answer (D): The sum of $4$ consecutive odd integers is always a multiple of $8$,
\[
(2n-3)+(2n-1)+(2n+1)+(2n+3)=8n.
\]
Among the given choices, only $100$ is not a multiple of $8$. The other four numbers can each be written as the sum of four consecutive odd numbers:
$16=1+3+5+7$
$40=7+9+11+13$
$72=15+17+19+21$
$20=47+49+51+53$
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