题目信息
An "Armstrong number" is an n-digit number that is equal to the sum of the nth powers of its individual digits. For example, 153 is an Armstrong number because it has 3 digits and 13 + 53 + 33 = 153. What is the digit k in the Armstrong number 1,6k 4 ?
A:2
B:3
C:4
D:5
E:6
参考答案及共享解析
共享解析来源为网络权威资源、GMAT高分考生等； 如有疑问，欢迎在评论区提问与讨论
本题耗时：
已选答案：
正确答案：
B:3
Arithmetic Operations with integers
If k = 1, then 1,6k4 = 1,614 and 14 + 64 + 14 + 44 is equal to 1 + 1,296 + 1 + 256 = 1,554 ≠ 1,614. Therefore, k = 1 does not produce an Armstrong number.
If k = 2, then 1,6k4 = 1,624 and 14 + 64 + 24 + 44 is equal to 1 + 1,296 + 16 + 256 = 1,569 ≠ 1,624. Therefore, k = 2 does not produce an Armstrong number.
If k = 3, then 1,6k4 = 1,634 and 14 + 64 + 34 + 44 is equal to 1 + 1,296 + 81 + 256 = 1,634. Therefore, k = 3 produces an Armstrong number.
Alternatively, the condition that 1,6k4 is an Armstrong number can be expressed by the equation 14 + 64 + k4 + 44 = 1,000 + 600 + 10k + 4, or 1,553 + k4 = 1,604 + 10k. This can be rewritten as k(k3 − 10) = 51. Therefore, k must be a factor of 51, and 3 is the only answer choice that is a factor of 51.
The correct answer is B.
If k = 1, then 1,6k4 = 1,614 and 14 + 64 + 14 + 44 is equal to 1 + 1,296 + 1 + 256 = 1,554 ≠ 1,614. Therefore, k = 1 does not produce an Armstrong number.
If k = 2, then 1,6k4 = 1,624 and 14 + 64 + 24 + 44 is equal to 1 + 1,296 + 16 + 256 = 1,569 ≠ 1,624. Therefore, k = 2 does not produce an Armstrong number.
If k = 3, then 1,6k4 = 1,634 and 14 + 64 + 34 + 44 is equal to 1 + 1,296 + 81 + 256 = 1,634. Therefore, k = 3 produces an Armstrong number.
Alternatively, the condition that 1,6k4 is an Armstrong number can be expressed by the equation 14 + 64 + k4 + 44 = 1,000 + 600 + 10k + 4, or 1,553 + k4 = 1,604 + 10k. This can be rewritten as k(k3 − 10) = 51. Therefore, k must be a factor of 51, and 3 is the only answer choice that is a factor of 51.
The correct answer is B.
加入收藏
在线答疑
题目来源
