Is a positive integer x odd?

Question

GMAT CLUB'S FRESH QUESTION

(1) The greatest common factor of the smallest positive factor of x and the largest positive factor of x is odd
(2) The least common multiple of the smallest positive factor of x and the largest positive factor of x is even

M36-26

EgmatQuantExpert · Answer

Solution

Given:  
• x is a positive integer

To find:  
• Is the value of x odd?

Approach and Working:   
As x is a positive integer,
• The smallest positive factor of x = 1
• The largest positive factor of x = x

Analysing Statement 1  
• As per the information provided in Statement 1, the GCF of the smallest positive factor of x and the largest positive factor of x is odd
 o Or, GCF (1, x) = odd 
• We know, the GCF of 1 and any other positive number will always be equal to 1 only, which is an odd number
 o Hence, we cannot say anything about the nature of the number x 
Hence, Statement 1 is not sufficient to answer

Analysing Statement 2  
• As per the information provided in Statement 2, the LCM of the smallest positive factor of x and the largest positive factor of x is even
 o Or, LCM (1, x) = even 
• We know, the LCM of 1 and any other positive number will always be equal to that number only – given that it is an even number
 o Hence, we can say that the number x is an even number 
Hence, Statement 2 is sufficient to answer

Hence, the correct answer is option B.

Answer: B

Bunuel · Answer

Par of  GMAT CLUB'S New Year's Quantitative Challenge Set

GMATin · Answer

Interesting question because it tests the basics of the topic LCM & HCF.

Two concepts that are tested are: 1) Smallest positive factor of a positive integer will always be 1 and 2) largest positive factor of a positive integer will always be the number itself

Statement 1: Given that GCF of 1 and x is odd, does not tell us anything about the number x. This is because whether x is odd or even, GCF between the number and 1 will always be 1 (i.e. odd). Hence,  this statement is insufficient

Statement 2: Given that LCM of 1 and x is even shows that x is definitely even. This is because LCM between a number and 1 is essentially x*1=x. Since this LCM is even, it proves that x is even.
Hence,  this statement is insufficient

Hence, answer is B

Bunuel · Answer

Official Solution:

Is a positive integer  x  odd? 
 
(1) The greatest common factor of the smallest positive factor of  x  and the largest positive factor of  x  is odd
 
The smallest positive factor of any positive integer is 1 and the largest positive factor of any positive integer is the integer itself. So, we are given that the greatest common factor of 1 and  x  is odd. This is true for any positive integer  x  because the greatest common factor of 1 and  x  is always 1, which is odd. So,  x  could be any positive even or any positive odd number. Not sufficient.
 
(2) The least common multiple of the smallest positive factor of  x  and the largest positive factor of  x  is even
 
The smallest positive factor of any positive integer is 1 and the largest positive factor of any positive integer is the integer itself. So, we are given that the least common multiple 1 and  x  is even. Since the least common multiple 1 and  x  is  x  itself, then we are basically told that  x  is even. Sufficient.

Answer: B

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Is a positive integer x odd?

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