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19 Jul 2024, 07:00
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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
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19 Jul 2024, 07:16
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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?
If n = 1; k = 1000 - 230 = 770 ; Sum of digits = 7+7+0 = 14
If n = 2; k = 1000000 - 230 = 999770; Sum of digits = (9+9+9) + 7 + 7 + 0 = 27 + 14 = 41
If n = 3; k = 1000000000 - 230 = 999999770; Sum of digits = (9+9+9) + (9+9+9) + 7 + 7 + 0 = 27*2 + 14 = 54 + 14 = 68
In general, If \(k = 1000^n - 230\); Sum of digits = 27(n-1) + 14 = 230
27n - 27 + 14 = 230
27n = 230 + 13 = 243
n = 243/27 = 9
IMO B
If n = 1; k = 1000 - 230 = 770 ; Sum of digits = 7+7+0 = 14
If n = 2; k = 1000000 - 230 = 999770; Sum of digits = (9+9+9) + 7 + 7 + 0 = 27 + 14 = 41
If n = 3; k = 1000000000 - 230 = 999999770; Sum of digits = (9+9+9) + (9+9+9) + 7 + 7 + 0 = 27*2 + 14 = 54 + 14 = 68
In general, If \(k = 1000^n - 230\); Sum of digits = 27(n-1) + 14 = 230
27n - 27 + 14 = 230
27n = 230 + 13 = 243
n = 243/27 = 9
IMO B
19 Jul 2024, 08:15
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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
Rewrite such that the base is 10 for simplification:
\(10^{3n}\) 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:
Answer: B.
GMAT Club Official Explanation:
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:
\(k = 1,000^n - 230 = \)
\(= 10^{3n} - 230\)
\(= 10^{3n} - 230\)
\(10^{3n}\) 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
3n - 3 = 24
n = 9
Answer: B.
