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19 Jul 2024, 07:00
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B
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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, 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.
General Discussion
19 Jul 2024, 07:14
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Given the equation k = (1000^n) - 230, where n is a positive integer, and the sum of the digits of k is 230, we need to find the value of n.
Step-by-Step Analysis:
1. Expression of k:
k = (1000^n) - 230
2. Simplify 1000^n:
1000^n = (10^3)^n = 10^(3n)
So,
k = 10^(3n) - 230
3. Structure of k:
For large values of n, 10^(3n) will be a number with 1 followed by 3n zeros. Subtracting 230 from this number means subtracting 230 from a number like 1000, 1000000, 1000000000, ..., which would result in a number that ends with the digits 770.
4. Example Calculation:
Let's consider smaller n to understand the pattern:
- For n = 1:
1000^1 - 230 = 1000 - 230 = 770
Sum of the digits: 7 + 7 + 0 = 14
- For n = 2:
1000^2 - 230 = 1000000 - 230 = 999770
Sum of the digits: 9 + 9 + 9 + 7 + 7 + 0 = 41
- For n = 3:
1000^3 - 230 = 1000000000 - 230 = 999999770
Sum of the digits: 9 + 9 + 9 + 9 + 9 + 9 + 7 + 7 + 0 = 68
Continuing this pattern, we see that each 1000^n adds three more nines and three more digits to the sum.
5. General Pattern:
Each additional 1000^n after the initial 10^(3n) contributes mainly with 9s added and the subtraction impacts at the end with the subtraction of 230 causing the terminal end digits be impacted less significantly.
Given requirement:
Sum of digits = 230
If 1000^n - 230 results in numbers where their summation of digits summing to 230 after the transformation which predominately following above pattern nines can contribute most parts.
By the pattern observed, significant additional figures required around 3 * (n - 1) + 7 to contribute to these transforming sums of:
Checking:
n = 9: 3 * 9 = 27
Since each digit adds contributing digits - such as total figures - each `9`s continuing sums.
Exact Solution for given:
n = 9
Conclusion:
The value of n that satisfies the given conditions is n = 9.
IMO B.
Step-by-Step Analysis:
1. Expression of k:
k = (1000^n) - 230
2. Simplify 1000^n:
1000^n = (10^3)^n = 10^(3n)
So,
k = 10^(3n) - 230
3. Structure of k:
For large values of n, 10^(3n) will be a number with 1 followed by 3n zeros. Subtracting 230 from this number means subtracting 230 from a number like 1000, 1000000, 1000000000, ..., which would result in a number that ends with the digits 770.
4. Example Calculation:
Let's consider smaller n to understand the pattern:
- For n = 1:
1000^1 - 230 = 1000 - 230 = 770
Sum of the digits: 7 + 7 + 0 = 14
- For n = 2:
1000^2 - 230 = 1000000 - 230 = 999770
Sum of the digits: 9 + 9 + 9 + 7 + 7 + 0 = 41
- For n = 3:
1000^3 - 230 = 1000000000 - 230 = 999999770
Sum of the digits: 9 + 9 + 9 + 9 + 9 + 9 + 7 + 7 + 0 = 68
Continuing this pattern, we see that each 1000^n adds three more nines and three more digits to the sum.
5. General Pattern:
Each additional 1000^n after the initial 10^(3n) contributes mainly with 9s added and the subtraction impacts at the end with the subtraction of 230 causing the terminal end digits be impacted less significantly.
Given requirement:
Sum of digits = 230
If 1000^n - 230 results in numbers where their summation of digits summing to 230 after the transformation which predominately following above pattern nines can contribute most parts.
By the pattern observed, significant additional figures required around 3 * (n - 1) + 7 to contribute to these transforming sums of:
Checking:
n = 9: 3 * 9 = 27
Since each digit adds contributing digits - such as total figures - each `9`s continuing sums.
Exact Solution for given:
n = 9
Conclusion:
The value of n that satisfies the given conditions is n = 9.
IMO B.
