This GRE maths sample question is a quantitative comparison question in Number Properties. Concept tested: Computing number of factors of integers; finding the smallest positive integer with 7 factors.
Question 1 : N is the smallest positive integer that has 7 factors.
|Column A||Column B|
|Number of factors of √N||Number of factors of (N-2)|
The number of factors of number N is 7. 7 is a prime number.
Note: If a number N can be prime factorized as ap × bq, where 'a' and 'b' are prime factors of N, number of factors of N = (p + 1) (q + 1)
So, 7 = 1 × 7 is the only way to express 7 as a product of 2 numbers
Inference: p + 1 = 1 and q + 1 = 7 or p = 0 and q = 6
So, any number that has 7 factors will have p = 0 and q = 6.
i.e., the number will have only one prime factor.
2 is the smallest prime number. So, the smallest number that will have 7 factors, N = 26 = 64.
Compute the number of factors of √N
N = 64
√N = √64 = 8
Prime factorize 8, we get 8 = 23
Number of factors of 8 = (3 + 1) = 4
Value of column A is 4.
Compute the number of factors of (N – 2),
N = 64
N - 2 = 64 – 2 = 62
Prime factorize 62, we get 62 = 2 × 31
Number of factors of 62 = (1 + 1) × ( 1 + 1) = 2 × 2 = 4
Value of column B is 4.
Column A: Number of factors of √N = 4
Column B: Number of factors of (N – 2) = 4
Both columns are equal.
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