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The sum of all the digits of the positive integer \(q\) is equal to the three-digit number \(x13.\) If \(q = 10^n - 49\)

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by Vincen » Mon Feb 15, 2021 7:55 am

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The sum of all the digits of the positive integer \(q\) is equal to the three-digit number \(x13.\) If \(q = 10^n - 49,\) what is the value of \(n?\)

(A) 24
(B) 25
(C) 26
(D) 27
(E) 28

Answer: B

Source: Manhattan GMAT
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Re: The sum of all the digits of the positive integer \(q\) is equal to the three-digit number \(x13.\) If \(q = 10^n -

by swerve » Mon Feb 15, 2021 4:18 pm
Vincen wrote: ↑
Mon Feb 15, 2021 7:55 am
 The sum of all the digits of the positive integer \(q\) is equal to the three-digit number \(x13.\) If \(q = 10^n - 49,\) what is the value of \(n?\)

(A) 24
(B) 25
(C) 26
(D) 27
(E) 28

Answer: B

Source: Manhattan GMAT
\(10^2 - 49 = 51\)
\(10^3 - 49 = 951\)
\(10^4 - 49 = 9951\)

The sum of digits \(5\) and \(1\) is \(5 + 1 = 6.\) In order to get \(x13,\) we need to add a units digit \(7\) to \(6.\)

(A) \(10^{24}\) gives \(22\,9s = 9\cdot 22 =\) units digit \(8\)
(B) \(10^{25}\) gives \(23\,9s = 9\cdot 23 =\) units digit \(7\)

Therefore, B
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