A number N^2 has 35 factors. How many factors can N have? GMAT Problem Solving

Question - A number N^2 has 35 factors. How many factors can N have?

1. 6 or 10 factors
2. 8 or 14 factors
3. 10 or 16 factors
4. 12 or 18 factors
5. 14 or 20 factors

‘A number N^2 has 35 factors. How many factors can N have?' - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “ GMAT Official Guide 2021”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation

Approach Solution 1: 
To understand the approach, we will first see one example.
Let us take an example 12.
Factors of 12 are 2- 2time and 3- 1 time.
Total no of factor - (2+1) * (1+1) =6.
For 12*12 - it has 2 - 4times and 3 - 2 timed.
Total no of factor - 5*3=15
Now, let us consider case 1:
N=x.y
For N*N
We will be trying a reverse approach. 35 = 5*7.
Factors are x - 4times and y- 6times.
For N => x -2 times and y- 3 times.
Total no of factor is (2+1)*(3+1)=12
In the Case 2
If N^2 is raised to the power of 34.
N^2 will have 34 factors.
And hence for N we have 17 factors.
Hence total no of factor as 17+1 = 18
The factors will be 12 or 18. Hence, D is the correct answer.

Correct Answer: D

Approach Solution 2: 
Case 1: N^2 is composed by two different primes
A number with 35 factors = a^6*b^4 (to count the number of factors we must add one to the power and multiply them, for instance, (6+1)*(4+1) = 7*5 = 35 factors.
So N = a^3*b^2. The number of factors of N can be obtained in the same way that we obtained the 35 factors of N^2.
Number of factors of N =(3+1)*(2+1) = 12 factors
Case 2: N^2 is composed by one prime
Using the same rule to count factors mentioned above, N^2 = a^34
so N = a^17 then N has 18 factors
Hence, the answer is 12 or 18 making D the correct option.

Correct Answer: D

Approach Solution 3: 

This problem can be solved by considering 2 cases

Case 1, we see
N=x.y(For N*N)
So, while trying reverse approach 35 we get 5*7.
Factors are x - 4times and that of y- 6times.
So, in this case N will be x -2 times and y- 3 times.
Therefore the total number of factor is (2+1)*(3+1)=12

Case 2, we see
If we consider N^2 being raised to the power of 34.
N^2 will have 34 factors.
And hence for N we have 17 factors.
Hence total no of factor as 17+1 = 18
Correct Answer: D

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