p is a positive integer greater than 100. Is p divisible by 36?

Question

(1) There are 8 positive integers, including 1 and p, which divide p.
(2) The highest positive integer less than or equal to 100 that divides p is 75.

Bunuel · Answer

p is a positive integer greater than 100. Is p divisible by 36?

Notice that 36 = 2^2*3^2 and has (2 + 1)(2 + 1) = 9 factors.

(1) There are 8 positive integers, including 1 and p, which divide p.

The number of factors of p (8) is less than the number of factors of 36 (9), so p cannot be divisible by 36. Sufficient.

(2) The highest positive integer less than or equal to 100 that divides p is 75.

p is divisible by 75. So, p is divisible by 25. If p were also divisible by 36 = 2^2*3^2, then it would have been divisible by 4, so the highest positive integer less than or equal to 100 that divides p would have been 25*4 = 100, not 75. Therefore, p cannot be divisible by 36. Sufficient.

Answer: D.

gurmukh · Answer

From statement 1
P in order to he divisible by 36,it should be a multiple of 36 and 36 has nine factors so p should have more than 9 factors. Hence we can say p is not divisible by 36. Sufficient
From statement 2
P is not divisible by 4 because it is not divisible by 100 therefore it is not divisible by 36. Sufficient
Hence D is the answer.

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GMATWhizTeam · Answer

Solution 
 Step 1: Analyse Question Stem 
 • p > 100
• p is a positive integer.
• We need to find if p is divisible by 36.
 o Or, if  p = 2^2*3^2*k  where k is a positive integer.

Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE 
 Statement 1:  There are 8 positive integers, including 1 and p, which divide p.
 • According to this statement, p has total 8 positive factors.
• Now, total positive factors of  36 = 2^2*3^2 = (2+1)(2+1) = 9 
 o Had p been in the form of  p =2^2*3^2*k , number of factors p would have been more than or equal to 9. 
• Since, total number of positive factors of p < total number of positive factors of 36
 o Therefore, p is not divisible by 36.  
Hence, statement 1 is sufficient and we can eliminate answer Options B, C and E.

Statement 2:  The highest positive integer less than or equal to 100 that divides p is 75.
 • p is divisible by 75 but not divisible by 100.
 o It means p is in the form of,  p = 5^2*3*n , but not in the form of,  p = 5^2*2^2*m 
o Where n and m are positive integer. 
• Therefore, p cannot be divisible by 36 as  2^2  in 36 won’t cancel out. 
Hence, statement 2 is also sufficient and the correct answer is  Option D.

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p is a positive integer greater than 100. Is p divisible by 36?

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p is a positive integer greater than 100. Is p divisible by 36?

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