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). |

- Quantity A is greater
- Quantity B is greater
- The two quantities are equal
- The relationship cannot be determined from the information given

#### Explanatory Answer

Video Explanation will be added soon### 3 steps to the answer

##### Use these hints to get the answer

- Find the smallest positive integer that will have 7 factors.
- Find the number of factors of \\sqrt{N}).
- 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 a^{p} * b^{q}, 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 2^{6} = 64.

#### Step 2: Compute number of factors of \\sqrt{N})

\\sqrt{64}) = 8.

**Method**: 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)

Prime factorize 8: 8 = 2^{3}.

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 a^{p} * b^{q}, 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.