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12 Days of Christmas GMAT Competition - Day 2: If k is a positive 12 Dec 2023, 07:00
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12 Days of Christmas 🎅 GMAT Competition with Lots of Questions & Fun

If k is a positive integer, how many positive factors does k have?

(1) k/11 is an integer which does not have a factor p such that 1 < p < k/11.

(2) The number of positive factors of k is odd.

 


This question was provided by GMAT Club
for the 12 Days of Christmas Competition

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C
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12 Days of Christmas GMAT Competition - Day 2: If k is a positive 16 Dec 2023, 06:58
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GMAT Club Official Explanation:



If k is a positive integer, how many positive factors does k have?

(1) k/11 is an integer which does not have a factor p such that 1 < p < k/11.

This statement implies that k/11 is a prime number, as a prime number only has two factors: 1 and itself, and no other factors between 1 and the number itself. Therefore, k = 11*prime. If this prime is any prime other than 11, then k has (1 + 1)(1 + 1) = 4 factors. However, if this prime is 11, then k = 11^2, and the number of factors is (2 + 1) = 3. Not sufficient.

(2) The number of positive factors of k is odd.

Only squares of integers have an odd number of factors, while all other integers have an even number of factors. Therefore, this statement just implies that k = integer^2. Not sufficient.

(1)+(2) Since from (2), k must be the square of an integer, then from (1) it follows that it must be 11^2, giving the number of factors equal to 3. Sufficient.

Answer: C.
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12 Days of Christmas GMAT Competition - Day 2: If k is a positive 12 Dec 2023, 07:50
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Bunuel
12 Days of Christmas 🎅 GMAT Competition with Lots of Questions & Fun

If k is a positive integer, how many positive factors does k have?

(1) k/11 is an integer which does not have a factor p such that 1 < p < k.

(2) The number of positive factors of k is odd.

 


This question was provided by GMAT Club
for the 12 Days of Christmas Competition

Win $40,000 in prizes: Courses, Tests & more

 

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(1) k/11 is an integer which does not have a factor p such that 1 < p < k.

k/11 is a prime number.

Hence, k can be 11 * 11. In this case the number of factors = 3

or

k can be 11 * some other prime number. In this case the number of factors = 4

The statement alone is not sufficient.

(2) The number of positive factors of k is odd.

The statement alone is not sufficient. k can be any odd number.

Statement (1) + Statement (2)

The statements combined tell us the number of factor is odd. Out of two possibilities in Statement 1, we can eliminate the possibility in which the number of factor is 4. Hence, the number of factor is 3.

The statements combined are sufficient.

IMO C
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12 Days of Christmas GMAT Competition - Day 2: If k is a positive 12 Dec 2023, 08:01
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If k is a positive integer, how many positive factors does k have?

(1) k/11 is an integer which does not have a factor p such that 1 < p < k. --> meaning that K is prime number
so K/11 is an integer Hence K = 11 then K has 2 factors.which is 1 and 11

(2) The number of positive factors of k is odd.--> not enough might have 1 or 3 factors.

Hence A.
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12 Days of Christmas GMAT Competition - Day 2: If k is a positive 12 Dec 2023, 08:35
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A.

1. means k=11 (2 factors), for any other value e.g 22, factor of 22/11 will lie between 1 and k
SUFFICIENT
2. cannot tell anything unique about k
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12 Days of Christmas GMAT Competition - Day 2: If k is a positive 12 Dec 2023, 08:46
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(1) k/11 is an integer which does not have a factor p such that 1 < p < k.

K/11 is an integer and since it does not have a factor p which is 1 < p < k. This implies that K is a prime number and only has two factors. Sufficient

(2) The number of positive factors of k is odd.
K could be any number. Insufficient
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12 Days of Christmas GMAT Competition - Day 2: If k is a positive 12 Dec 2023, 09:13
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stmnt 1= it says k/11= x and no such factor p such that 1<p<x----means x = prime and also 11 cannot have power>2 since then no such x is possible
Still k can be (11)^2 (factors=3) or 11*2(factors=4)-----Not sufficient
stmnt 2= factors are odd maybe k is perfect square or may not be
possible value= 11^2 (factors=3) or 2^2*11^2(factors=9)----not sufficient
combining both k=11^2 --- (factors=3)
ans=C
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12 Days of Christmas GMAT Competition - Day 2: If k is a positive 12 Dec 2023, 09:50
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If k is a positive integer, how many positive factors does k have?

(1) k/11 is an integer which does not have a factor p such that 1 < p < k/11.
This implies that k/11 is a prime number
If the prime number is 2 - the number of positive factors in k(=22) are 4 (k = 2*11) : 1, 2, 11, 22
But if the prime number is 11 - the number of positive factors in k (=121) are 3 : 1, 11, 121
Not sufficient

(2) The number of positive factors of k is odd.
Statement 2 alone is not sufficient as it only implies that k is a perfect square.

statement 1 and 2 combined
k/11 is an integer which does not have a factor p such that 1 < p < k/11 => k = 121. As in this case, k/11 = 11 which is a prime number
The number of positive factors of k is odd => k is a perfect square
Hence, k = 121 has 3 positive factors : 1, 11, 121
Sufficient
Answer C
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12 Days of Christmas GMAT Competition - Day 2: If k is a positive 17 Dec 2023, 03:24
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Bunuel

GMAT Club Official Explanation:



If k is a positive integer, how many positive factors does k have?

(1) k/11 is an integer which does not have a factor p such that 1 < p < k/11.

This statement implies that k/11 is a prime number, as a prime number only has two factors: 1 and itself, and no other factors between 1 and the number itself. Therefore, k = 11*prime. If this prime is any prime other than 11, then k has (1 + 1)(1 + 1) = 4 factors. However, if this prime is 11, then k = 11^2, and the number of factors is (2 + 1) = 3. Not sufficient.

(2) The number of positive factors of k is odd.

Only squares of integers have an odd number of factors, while all other integers have an even number of factors. Therefore, this statement just implies that k = integer^2. Not sufficient.

(1)+(2) Since from (2), k must be the square of an integer, then from (1) it follows that it must be 11^2, giving the number of factors equal to 3. Sufficient.

Answer: C.
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Dear Bunuel, please help to clarify if my interpretation of the two statements is incorrect.

(1) k/11 is an integer which does not have a factor p such that 1 < p < k/11.

This implies 'k' doesn't have a prime factor less than 11. There can be other prime factors greater than 11. Therefore, this statement by itself is not sufficient.

(2) The number of positive factors of k is odd.

This implies all the prime factors of 'k' have even powers in the factorisation. We need to know the exact number of prime factors and the exact powers to which they are raised in the factorisation of integer 'k'. Therefore, this statement by itself is not sufficient.

Even after combining both statements, we still don't have the exact number of prime factors and the exact powers to which they are raised.

Shouldn't the answer be 'E' in this case?

Thanks!
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12 Days of Christmas GMAT Competition - Day 2: If k is a positive 17 Dec 2023, 05:23
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Bunuel

GMAT Club Official Explanation:



If k is a positive integer, how many positive factors does k have?

(1) k/11 is an integer which does not have a factor p such that 1 < p < k/11.

This statement implies that k/11 is a prime number, as a prime number only has two factors: 1 and itself, and no other factors between 1 and the number itself. Therefore, k = 11*prime. If this prime is any prime other than 11, then k has (1 + 1)(1 + 1) = 4 factors. However, if this prime is 11, then k = 11^2, and the number of factors is (2 + 1) = 3. Not sufficient.

(2) The number of positive factors of k is odd.

Only squares of integers have an odd number of factors, while all other integers have an even number of factors. Therefore, this statement just implies that k = integer^2. Not sufficient.

(1)+(2) Since from (2), k must be the square of an integer, then from (1) it follows that it must be 11^2, giving the number of factors equal to 3. Sufficient.

Answer: C.

Dear Bunuel, please help to clarify if my interpretation of the two statements is incorrect.

(1) k/11 is an integer which does not have a factor p such that 1 < p < k/11.

This implies 'k' doesn't have a prime factor less than 11. There can be other prime factors greater than 11. Therefore, this statement by itself is not sufficient.

(2) The number of positive factors of k is odd.

This implies all the prime factors of 'k' have even powers in the factorisation. We need to know the exact number of prime factors and the exact powers to which they are raised in the factorisation of integer 'k'. Therefore, this statement by itself is not sufficient.

Even after combining both statements, we still don't have the exact number of prime factors and the exact powers to which they are raised.

Shouldn't the answer be 'E' in this case?

Thanks!
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For the answer to be E, we should have two or more answers to the question "how many positive factors does k have?". Can you please give examples of that? Can you please give any other value of k other than 11^2 that satisfies both statements?
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12 Days of Christmas GMAT Competition - Day 2: If k is a positive 07 Apr 2025, 09:45
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