AMC8 2018
AMC8 2018 · Q25
AMC8 2018 · Q25. It mainly tests Perfect squares & cubes.
How many perfect cubes lie between $2^8 + 1$ and $2^{18} + 1$, inclusive?
有多少个完全立方数位于$2^8 + 1$和$2^{18} + 1$之间（包含边界）？
(A)
4
4
(B)
9
9
(C)
10
10
(D)
57
57
(E)
58
58
Answer
Correct choice: (E)
正确答案：（E）
Solution
Answer (E): Note that $2^8+1=257$ and that $216=6^3<257<7^3=343$. Also note that $2^{18}=(2^6)^3=64^3$, so the perfect cube that is closest to and less than $2^{18}+1$ is $64^3$. Thus the numbers $7^3$, $8^3$, $\ldots$, $63^3$, $64^3$ are precisely the perfect cubes that lie between the two given numbers, and so there are $64-6=58$ of these perfect cubes.
答案（E）：注意到 $2^8+1=257$，并且 $216=6^3<257<7^3=343$。还注意到 $2^{18}=(2^6)^3=64^3$，因此小于且最接近 $2^{18}+1$ 的完全立方数是 $64^3$。所以 $7^3$、$8^3$、$\ldots$、$63^3$、$64^3$ 恰好是位于给定两个数之间的完全立方数，因此这些完全立方数共有 $64-6=58$ 个。
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