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) |

- Column A is greater
- Column B is greater
- The two quantities are equal
- Cannot be determined

From INR 2999

The number of factors of number N is 7. 7 is a prime number.

**Note**: If a number N can be prime factorized as a^{p} × b^{q}, 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 = 2^{6} = 64.

**Compute the number of factors of √N**

N = 64

√N = √64 = 8

Prime factorize 8, we get 8 = 2^{3}

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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