# GRE® Number Properties Practice

Concept: Number of factors of an integer.

This GRE quant practice question is a number properties quantitative comparison question. Concept tested: Computing number of factors of integers; using that concept finding the smallest positive integer with 7 factors.

#### Question: N is the smallest positive integer that has 7 factors.

Quantity A Quantity B
Number of factors of √N. Number of factors of (N-2).
1. Quantity A is greater
2. Quantity B is greater
3. The two quantities are equal
4. The relationship cannot be determined from the information given

Video Explanation will be added soon

### 3 steps to the answer

##### Use these hints to get the answer
1. Find the smallest positive integer that will have 7 factors.
2. Find the number of factors of $$sqrt{N}$. 3. Find the number of factors of$N - 2).

### Evaluate Quantity A

#### Step 1: Find the smallest positive integer that has 7 factors

The number of factors, 7, is a prime number. ∴ the only way to express 7 as a product of 2 numbers is 1 * 7.
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, 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 is 26 = 64.

#### Step 2: Compute number of factors of $$sqrt{N}$ $\sqrt{64}$ = 8. Method: 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)
Prime factorize 8: 8 = 23.
Number of factors of 8 = (3 + 1) = 4.

Value of Quantity A is 4.

### Evaluate Quantity B

#### Compute number of factors of (N - 2)

N = 64. Therefore, (N - 2) = 62
Method: 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)
Prime factorize 62: 62 = 2 * 31
Number of factors of 62 = (1 + 1) * (1 + 1) = 4

Value of Quantity B is 4.

### The Comparison

Quantity A: Number of factors of $$sqrt{N}$ = 4 Quantity B: Number of factors of$N - 2) = 4
Both quantities are equal. Choice C is the answer.