Number Theory Questions - Number of Factors

Two more questions in Number Theory. These are focused on finding the number of factors of a given number.

Question 1
Numbers A, B, C and D have 16, 28, 30 and 27 factors respectively. Which of these could be a perfect cube?

Question 2 
If a three digit number  of the form 'abc' has 3 factors (where a, b and c stand for three digits), how many factors does the 6-digit number 'abcabc' have?

Correct Answers:
Question 1: A and B could be perfect cubes
Question 2: 16 factors or 24 factors

Explanations

1. Numbers A, B, C and D have 16, 28, 30 and 27 factors. Which of these could be a perfect cube?

Any number of the form p^aq^br^c will have (a+1) (b+1)(c+1) factors, where p, q, r are prime. In order for the number to be a perfect cube a, b, c will have to be multiples of 3.

We can assume that a = 3m, b = 3n, c = 3l

This tells us that the number of factors will have to be of the form (3n+1) *(3m+1) *(3l+1). In other words (a +1), (b + 1) and (c + 1) all leave a remainder of 1 on division by 3. So, the product of these three numbers should also leave a remainder of 1 on division by 3. Of the four numbers provided, 16 and 28 can be written in this form, the other two cannot.

So, a perfect cube can have 16 or 28 factors. Now, let us think about what kind of numbers will have 16 factors.

A number of the form p¹⁵ or q³r³ will have exactly 16 factors. Both are perfect cubes. Note that there are other prime factorizations possible that can have exactly 16 factors. But these two forms are perfect cubes, which is what we are interested in

Similarly, a number of the form p²⁷ or q³r⁶ will have 28 factors. Both are perfect cubes.

Given the number of factors, one should be able to write down the basic prime factorization forms that could lead to these many factors. This is a critical skillset for tackling some of the tougher questions in CAT.

2. If a three digit number ‘abc’ has 3 factors, how many factors does the 6-digit number ‘abcabc’ have?

‘abc’ has exactly 3 factors, so ‘abc’ should be square of a prime number. (This is an important inference, please remember this).

Any number of the form p^aq^br^c will have (a+1) (b+1)(c+1) factors, where p, q, r are prime. So, if a number has 3 factors, its prime factorization has to be p²

‘abcabc’ = ‘abc’ *1001 or abc * 7*11*13 (again, this is a critical idea to remember)

Now, ‘abc’ has to be square of a prime number. It can be either 121 or 169 (square of either 11 or 13) or it can be the square of some other prime number

When abc = 121 or 169, then ‘abcabc’ is of the form p³q¹r¹ 1, which should have 4*2*2 = 16 factors

When ‘abc’ = square of any other prime number (say 17² which is 289) , then ‘abcabc’ is of the form p¹q¹r¹s² , which should have 2*2*2*3 = 24 factors

So, ‘abcabc’ will have either 16 factors or 24 factors