### 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?

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

^{a}q^{b}r^{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 p

^{15 }or q^{3}r^{3 }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 inSimilarly, a number of the form p

^{27 }or q

^{3}r

^{6 }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

^{a}q^{b}r^{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 p^{2}
‘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 p

^{3}q^{1}r^{1 }1, which should have 4*2*2 = 16 factors
When ‘abc’ = square of any other prime number (say 17

^{2}which is 289) , then ‘abcabc’ is of the form p^{1}q^{1}r^{1}s^{2 }, which should have 2*2*2*3 = 24 factors
So, ‘abcabc’ will have either 16 factors or 24 factors

Labels: CAT number theory, CAT questions

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