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29 Jul 2024, 05:09
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If \(k = 1,000^n - 230\), where \(n\) is a positive integer, and the sum of the digits of \(k\) is 230, what is the value of \(n\)?
A. 8
B. 9
C. 10
D. 24
E. 27
A. 8
B. 9
C. 10
D. 24
E. 27
29 Jul 2024, 05:09
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Official Solution:
If \(k = 1,000^n - 230\), where \(n\) is a positive integer, and the sum of the digits of \(k\) is 230, what is the value of \(n\)?
A. 8
B. 9
C. 10
D. 24
E. 27
Rewrite such that the base is 10 for simplification:
\(k = 1,000^n - 230 = \)
\(= 10^{3n} - 230\)
\(10^{3n}\) will have \(3n + 1\) digits, 1 followed by \(3n\) zeros, the same way for example as \(10^4 = 10,000\) has \(4 + 1 = 5\) digits, 1 followed by 4 zeros. Then, \(10^{3n} - 230\) will have one less, so \(3n\) digits, comprising \(3n - 3\) nines followed by 770 at the end. The same way for example as \(10^4 = 10,000 - 230 = 9,770\) has \(4 - 3 = 1\) nine, followed by 770.
Given that the sum of the digits of \(k\) is 230, we'd have \((3n - 3)*9 + 7 + 7 + 0 = 230\):
\((3n - 3)*9 = 216\)
\(3n - 3 = 24\)
\(n = 9\)
Answer: B
If \(k = 1,000^n - 230\), where \(n\) is a positive integer, and the sum of the digits of \(k\) is 230, what is the value of \(n\)?
A. 8
B. 9
C. 10
D. 24
E. 27
Rewrite such that the base is 10 for simplification:
\(k = 1,000^n - 230 = \)
\(= 10^{3n} - 230\)
\(10^{3n}\) will have \(3n + 1\) digits, 1 followed by \(3n\) zeros, the same way for example as \(10^4 = 10,000\) has \(4 + 1 = 5\) digits, 1 followed by 4 zeros. Then, \(10^{3n} - 230\) will have one less, so \(3n\) digits, comprising \(3n - 3\) nines followed by 770 at the end. The same way for example as \(10^4 = 10,000 - 230 = 9,770\) has \(4 - 3 = 1\) nine, followed by 770.
Given that the sum of the digits of \(k\) is 230, we'd have \((3n - 3)*9 + 7 + 7 + 0 = 230\):
\((3n - 3)*9 = 216\)
\(3n - 3 = 24\)
\(n = 9\)
Answer: B
30 Sep 2024, 20:31
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Lovely question
