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.
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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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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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