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Question : (If x is an integer, then how many digits does \(X^{101}\) have?)

(1) \(X^{7}\) has 15 digits. (2) The units digits of \(X^{74}\) is 1

Its not difficult rather time consuming and not a typical gmat question.

1. x could be any number between 100 and 138 (with the help of computer) 2. x could be any integer/number ending unit digit in 1 and 9 (not 3 and 7).

Togather, x could be either of 101, 109 or 111 or 119 or 121 or 129 or 131. They do not have same number of digits if each is raised to power 101. So E.
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Question : (If x is an integer, then how many digits does \(X^{101}\) have?)

(1) \(X^{7}\) has 15 digits.

(2) The units digits of \(X^{74}\) is 1

(1) 100<=x< (1.d)*100 (where (1.d)^7 must be <10)

(1.d)^7 <10 ---> (1.d)^101 may be >10 --> which leads to multiple solutions.

not sufficient

(2) last digit is either 1 or 9 not sufficient

combined.

x can be 101,109,111,121.. etc. e.g 101= 1.01 *100 power 7 leads to 15 digits. --> power 101 leads N number of digits 111= 1.11 *100 power 7 leads to 15 digits.--> power 101 leads more than N number of digits

Will go with E.
_________________

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Question : (If x is an integer, then how many digits does \(X^{101}\) have?)

(1) \(X^{7}\) has 15 digits.

(2) The units digits of \(X^{74}\) is 1

From Question Stem: Let k = x^101. log (k) = 101 log (x).

From Statement 1: let P= x^7. log (p) = 7 log (x).

Now given log(P) = 15, So log(x) = 15/7. Insert the value of log (x) in the question stem. we get k= 101* 15/7. The value of k is the no. of digits in x^101

Question : (If x is an integer, then how many digits does \(X^{101}\) have?)

(1) \(X^{7}\) has 15 digits.

(2) The units digits of \(X^{74}\) is 1

From Question Stem: Let k = x^101. log (k) = 101 log (x).

From Statement 1: let P= x^7. log (p) = 7 log (x).

Now given log(P) = 15, So log(x) = 15/7. Insert the value of log (x) in the question stem. we get k= 101* 15/7. The value of k is the no. of digits in x^101

So A is sufficient.

Please comment guys..!!

Unfortunately that doesn't quite work. If we're working in base 10, and p has fifteen digits, then that means that 10^14 < p < 10^15. In other words, that means that 14 < log(p) < 15. There's no way to find an exact value for log(p) here without more information.

(and so as not to cause any panic among test takers, logarithms are definitely *not* tested on the GMAT!)
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GMAT Tutor in Toronto

If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com

Question : (If x is an integer, then how many digits does \(X^{101}\) have?)

(1) \(X^{7}\) has 15 digits.

(2) The units digits of \(X^{74}\) is 1

From Question Stem: Let k = x^101. log (k) = 101 log (x).

From Statement 1: let P= x^7. log (p) = 7 log (x).

Now given log(P) = 15, So log(x) = 15/7. Insert the value of log (x) in the question stem. we get k= 101* 15/7. The value of k is the no. of digits in x^101

So A is sufficient.

Please comment guys..!!

Unfortunately that doesn't quite work. If we're working in base 10, and p has fifteen digits, then that means that 10^14 < p < 10^15. In other words, that means that 14 < log(p) < 15. There's no way to find an exact value for log(p) here without more information.

(and so as not to cause any panic among test takers, logarithms are definitely *not* tested on the GMAT!)

Hello Ian Stewart

I agree that we don't get any exact integer value of log(p). But just for knowledge can you tell whether the approach was right.

From Question Stem: Let k = x^101. log (k) = 101 log (x).

From Statement 1: let P= x^7. log (p) = 7 log (x).

Now given log(P) = 15, So log(x) = 15/7. Insert the value of log (x) in the question stem. we get k= 101* 15/7. The value of k is the no. of digits in x^101

So A is sufficient.

________

Hello Ian Stewart

I agree that we don't get any exact integer value of log(p). But just for knowledge can you tell whether the approach was right.

Yes, you can use your method to determine precisely what Statement 1 tells us, provided that you use the correct inequalities - it's probably the best approach here, at least without a calculator (please note that this is *not* tested on the GMAT - for interest only!).

from which you can see that the number of digits in x^101 could range from 203 to 217.
_________________

GMAT Tutor in Toronto

If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com

what is the OA and OE for this ? is this a gmat question

I take it, from the comments above, that this is a question designed by one of the members here. It is not a real GMAT question, and you won't see something quite like this on test day - I suppose there's a small chance you could see a much simpler question testing the same concepts, but not with numbers nearly as large.
_________________

GMAT Tutor in Toronto

If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com

I believe the answer is E as well... my technique is not nearly as refined as the one posted by Ian above though:

Statement 2: The Units digit of \(X^{74}\) is 1.

i.e X is an integer with a units digit of 1 or 9. Not sufficient.

Statement 1: \(X^7\) has 15 digits.

\(10^{14} < X^7 < 10^{15}\)

\(100 < X < 100*r\)

...where r = \(\sqrt[7]{10}\)

Also, since we know X is greater than or equal to 100, and we are taking X^101, we need to know the EXACT value of X in order to determine how many digits X^101 has since:

\(\frac{101^{101}}{100^{101}} = 1.01^{101} > 10\)

i.e. X^101 and (X+1)^{101} will always differ by a factor of at least 10, meaning they will never have the same amount of digits.

Now here's where it gets kind of messy:

I know 1.3^7 = 1.69 * 1.69 * 1.69 * 1.3 < 1.7^3 * 1.3 < 10 So I know that the upper bound is AT LEAST 130 on X.

Therefore, insufficient.

Evaluating Both Statements Together:

Statement 2: X has units digit of 1 or 9 Statement 1: 100 <= X <= 130 (conservative estimate), must know X to exact number

101, 109, 111, 119... etc. there are multiple numbers that satisfy both cases and we know that they will all have different amount of digits. Therefore, insufficient.

Therefore, the correct answer is E. Sorry if this is confusing to some, I don't know an elegant solution to this problem other than the logarithmic one produced above.

ans shld be e coz 100^7 = 10^14 so from here numbering of 15 digits starts ... but it could be 101, 109, 111 all have 15 digits b statement nothing helps ... so ans e

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