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If a positive integer n has k positive factors, 2n has 2k positive 25 Sep 2023, 01:50
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

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95% (hard)

Question Stats:

43% (03:10) correct 57% (02:50) wrong based on 161 sessions

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If a positive integer n has k positive factors, 2n has 2k positive factors, and 6n has 3k positive factors, which of the following can be the value of n?

I. 177
II. 231
III. 279

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III
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C
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If a positive integer n has k positive factors, 2n has 2k positive 25 Sep 2023, 08:12
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Bunuel
If a positive integer n has k positive factors, 2n has 2k positive factors, and 3n has 6k positive factors, which of the following can be the value of n?

I. 177
II. 231
III. 279

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III
Show more

If a positive integer n has k positive factors, 2n has 2k positive factors,

i.e. n = ODD. But all numbers here are odd

n = 177 = 3*59
Number of factors of 177 = (1+1)*(1+1) = 4
3n = 3*177 = 3^2*59
Number of factors of 3*177 = (2+1)*(1+1) = 6
so the factors are not 6 times OUT


n = 231 = 3*7*11
Number of factors of 231 = (1+1)*(1+1)*(1+1) = 8
3n = 3*231 = 3^2*7*11
Number of factors of 3*231 = (2+1)*(1+1)*(1+1) = 12
so the factors are not 6 times OUT


n = 279 = 3^2*31
Number of factors of 279 = (2+1)*(1+1) = 6
3n = 3*279 = 3^3*31
Number of factors of 3*279 = (3+1)*(1+1) = 8
so the factors are not 6 times OUT


No value can be n as per second condition.
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If a positive integer n has k positive factors, 2n has 2k positive 25 Sep 2023, 09:33
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Bunuel
If a positive integer n has k positive factors, 2n has 2k positive factors, and 3n has 6k positive factors, which of the following can be the value of n?

I. 177
II. 231
III. 279

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III

If a positive integer n has k positive factors, 2n has 2k positive factors,

i.e. n = ODD. But all numbers here are odd

n = 177 = 3*59
Number of factors of 177 = (1+1)*(1+1) = 4
3n = 3*177 = 3^2*59
Number of factors of 3*177 = (2+1)*(1+1) = 6
so the factors are not 6 times OUT


n = 231 = 3*7*11
Number of factors of 231 = (1+1)*(1+1)*(1+1) = 8
3n = 3*231 = 3^2*7*11
Number of factors of 3*231 = (2+1)*(1+1)*(1+1) = 12
so the factors are not 6 times OUT


n = 279 = 3^2*31
Number of factors of 279 = (2+1)*(1+1) = 6
3n = 3*279 = 3^3*31
Number of factors of 3*279 = (3+1)*(1+1) = 8
so the factors are not 6 times OUT


No value can be n as per second condition.
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Sorry, there was a typo in the text. Edited. Can you please try again? Thank you!
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If a positive integer n has k positive factors, 2n has 2k positive 25 Sep 2023, 11:30
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177 = 3^1 x 59^1; Factors = 2 x 2 (add 1 to each power and mulitply) = 4 =k
177 x 2 = 2^1 x 59 ^1 x 3^1 = 2x2x2 = 8 = 2k
177 x 6 = 2^1 x 59^1 x 3^2 = 2x2x3 = 12 = 3k

Similarly 231 also follows the same; however, 279 doesnt.

Therefore, Ans = C
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If a positive integer n has k positive factors, 2n has 2k positive 25 Sep 2023, 11:40
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Use option to solve
n is 177 factors (k)= 3*59;4
2*177 factors 8 i.e. 2k
And 6n factors
2*3^2*59: 12=3k
Sufficient

If n is 231
3*7*11 :8
2*231:16 factors i.e. 2k
And 6n has
2*3^2*7*11 has 24 i.e. 3k
Sufficient

If n is 279
K here is 3^2*31= 6
2k has 2*3^2*31= 12=2k
6k has 2*3^3*31= 16 not equal to 3k
InSufficient

OPTION C IS CORRECT


Bunuel
If a positive integer n has k positive factors, 2n has 2k positive factors, and 6n has 3k positive factors, which of the following can be the value of n?

I. 177
II. 231
III. 279

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III
Show more

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If a positive integer n has k positive factors, 2n has 2k positive 25 Sep 2023, 22:52
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Bunuel
If a positive integer n has k positive factors, 2n has 2k positive factors, and 6n has 3k positive factors, which of the following can be the value of n?

I. 177
II. 231
III. 279

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III
Show more

We can find the number of factors for 2n, and 3n as shown below -

Attachment:
Screenshot 2023-09-26 111948.jpg
Screenshot 2023-09-26 111948.jpg [ 80.5 KiB | Viewed 4677 times ]

For I and II the positive integer n has k positive factors, 2n has 2k positive factors, and 6n has 3k positive factors.

Option C
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If a positive integer n has k positive factors, 2n has 2k positive 26 Sep 2023, 06:36
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Bunuel
If a positive integer n has k positive factors, 2n has 2k positive factors, and 6n has 3k positive factors, which of the following can be the value of n?

I. 177
II. 231
III. 279

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III
Show more

I. 177
\(n=177=59*3\)
k\(=(1+1)*(1+1) = 4\)
\(2n=59*3*2\)
\(k=2*2*2=8=2\)k
\(6n=59*3^{2}*2\)
\(k=2*3*2=12=3\)k

So 177 can be the value of n

II. 231
\(n=231=3*7*11\)
k\(=2*2*2=8\)
\(2n=2*3*7*11\)
\(k=2*2*2*2=16=2\)k
\(6n=2*3^{2}*7*11\)
\(k=2*3*2*2=24=3\)k

231 can be the value of n

III. 279

\(n=279=3^{2}*31\)
k=\(3*2=6\)
\(2n=2*3^{2}*31\)
\(k=2*3*2=12=2\)k
\(6n=2*3^{3}*31\)
\(k=2*4*2=16\) does not equal 3k
279 can not be the value of n

The correct answer is C
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If a positive integer n has k positive factors, 2n has 2k positive 23 Jul 2024, 07:46
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Bunuel
If a positive integer n has k positive factors, 2n has 2k positive factors, and 6n has 3k positive factors, which of the following can be the value of n?

I. 177
II. 231
III. 279

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III
Show more
If you observe the conditions carefully you will notice that when "a positive integer n has k positive factors, 2n has 2k positive factors." This shows n isnt multiple of 2.
Similarly when you multiply n with 6(2*3) then it is changing into 3k means the 1 power of 3 has to be there. (Try it by taking prime numbers. like take 3*7 as n perform the similar operations as given you will notice the pattern)
­n should contain zero power of 2 and only one power of 3 each those who satisifies this conditions are possible values of n.
In this case only I and II follows
Answer is C.
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If a positive integer n has k positive factors, 2n has 2k positive 23 Jul 2024, 07:53
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If a positive integer n has k positive factors, 2n has 2k positive factors, and 6n has 3k positive factors, which of the following can be the value of n?

I. 177 = 3*59; Number of positive factors k = 2*2=4; 2n = 2*3*59; Number of positive factors = 2*2*2 = 8 = 2k; 6n = 2*3^2*59; Number of positive factors = 2*3*2 = 12 = 3k; Possible
II. 231 = 3*7*11; Number of positive factors k = 2*2*2 = 8; 2n = 2*3*7*11; Number of positive factors = 2*2*2*2 = 16 = 2k; 6n = 2*3^2*7*11; Number of positive factors = 2*3*2*2 = 24 = 3k; Possible
III. 279 = 3^2*31; Number of positive factors k = 3*2 = 6; 2n = 2*3^2*31; Number of positive factors = 2*3*2 = 12 = 2k; 6n = 2*3^3*31; Number of positive factors = 2*4*2 = 16 <> 3k; Not Possible

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III

IMO C
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