2. 外语
  3. 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 i

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 i

admin2022-10-18  64
问题 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?
选项 A、2
B、3
C、4
D、5
E、6
答案B
解析 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 12 + 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.
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