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If x is a k-digit integer and x =(4^9)*(5^17), what is the value of k?

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corrayatom
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If x is a k-digit integer and x =(4^9)*(5^17), what is the value of k? [#permalink] New post   Updated on: 16 Apr 2024, 11:11
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If x is a k-digit integer and x = (4^9)(5^17), what is the value of k?

(A) 16
(B) 17
(C) 18
(D) 19
(E) 20
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C

Originally posted by corrayatom on 20 Jun 2023, 06:04.
Last edited by Bunuel on 16 Apr 2024, 11:11, edited 1 time in total.
Edited the options.
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Re: If x is a k-digit integer and x =(4^9)*(5^17), what is the value of k? [#permalink] New post   20 Jun 2023, 07:54
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corrayatom wrote:
If x is a k-digit integer and x = (4^9)(5^17), what is the value of k?

A. 9
B. 15
C. 16
D. 17
E. 18

\(x = (4^9)(5^{17})\)

\(x = (2^{18})(5^{17})\)

Or, \(x = (10^{17})2\)

\(10^{17}\) will have \(17\) ,\(0\)'s multiplied by 2 will have 1 more digit, ie 18 digits, Answer must be (E) 18
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Re: If x is a k-digit integer and x =(4^9)*(5^17), what is the value of k? [#permalink] New post   29 Jun 2023, 22:32
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houston1980 wrote:
If x is a k-digit integer and x = \(4^{9}*5^{17}\), what is the value of k?

(A) 16
(B) 17
(C) 18
(D) 19
(E) 20


x = \(4^{9}*5^{17}\)

x = \(2^{18}*5^{17}\)

x = \(2^{1}*10^{17}\)

Let's take a smaller value of the power to understand the pattern

x = \(2^{1}*10^{2}\) = 200 ; in this case x is 3 digit integer.

x = \(2^{1}*10^{3}\) = 2000 ; in this case x is 4 digit integer.

Therefore the number of digits = value of the exponent + 1

In this case,

number of digits in x= 17 + 1 = 18

Option C
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Re: If x is a k-digit integer and x =(4^9)*(5^17), what is the value of k? [#permalink] New post   07 Jul 2023, 07:33
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houston1980 wrote:
If x is a k-digit integer and x = \(4^{9}*5^{17}\), what is the value of k?

(A) 16
(B) 17
(C) 18
(D) 19
(E) 20


x = (4^9)(5^17) = (2^18)(5^17) = (2)(2 × 5)^17

x = (2)(10)^17

Multiplying an integer by (10)^17 simply attaches 17 zeros to the last digit of the integer.

Since 2 is a one-digit integer, the number of digits in question is:

1 + 17 = 18

Answer: C
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Re: If x is a k-digit integer and x =(4^9)*(5^17), what is the value of k? [#permalink] New post   18 Feb 2024, 11:39
Abhishek009 wrote:
corrayatom wrote:
If x is a k-digit integer and x = (4^9)(5^17), what is the value of k?

A. 9
B. 15
C. 16
D. 17
E. 18

\(x = (4^9)(5^{17})\)

\(x = (2^{18})(5^{17})\)

Or, \(x = (10^{17})2\)

\(10^{17}\) will have \(17\) ,\(0\)'s multiplied by 2 will have 1 more digit, ie 18 digits, Answer must be (E) 18

­Hi Abhi, thank you for your approach. One question I have is why do we need to multiply 10^17*2...
The property says x^a * y^a = (x*y)^a
I understand the solving until 2^18 * 5^17 = 10^17 --> We are using approximation right? in order to have the same exponent.
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Re: If x is a k-digit integer and x =(4^9)*(5^17), what is the value of k? [#permalink] New post   18 Feb 2024, 11:49
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ruis wrote:
Abhishek009 wrote:
corrayatom wrote:
If x is a k-digit integer and x = (4^9)(5^17), what is the value of k?

A. 9
B. 15
C. 16
D. 17
E. 18

\(x = (4^9)(5^{17})\)

\(x = (2^{18})(5^{17})\)

Or, \(x = (10^{17})2\)

\(10^{17}\) will have \(17\) ,\(0\)'s multiplied by 2 will have 1 more digit, ie 18 digits, Answer must be (E) 18

­Hi Abhi, thank you for your approach. One question I have is why do we need to multiply 10^17*2...
The property says x^a * y^a = (x*y)^a
I understand the solving until 2^18 * 5^17 = 10^17 --> We are using approximation right? in order to have the same exponent.

­
No approximation there.

    \(x = (4^9)(5^{17})=\)

    \( = (2^{18})(5^{17})=\)

    \( = 2(2^{17})(5^{17})=\)

    \( = 2(10^{17})\)
 ­
So, x is 2 followed by 17 zeros (200,000,000,000,000,000), making it an 18-digit number.­

Answer: E.
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Re: If x is a k-digit integer and x =(4^9)*(5^17), what is the value of k? [#permalink] New post   18 Feb 2024, 12:01
That´s amazing, thank you so much Bunuel! You saved me a huge misunderstanding there
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Re: If x is a k-digit integer and x =(4^9)*(5^17), what is the value of k? [#permalink] New post   16 Apr 2024, 11:04
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­Bunuel

When I saw this question in the gmat focus the answer choices were

A. 16
B. 17
C. 18
D. 19
E. 20
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Re: If x is a k-digit integer and x =(4^9)*(5^17), what is the value of k? [#permalink] New post   16 Apr 2024, 11:11
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carcass wrote:
­Bunuel

When I saw this question in the gmat focus the answer choices were

A. 16
B. 17
C. 18
D. 19
E. 20

­Edited the options. Thank you!
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Re: If x is a k-digit integer and x =(4^9)*(5^17), what is the value of k? [#permalink]
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