CAT Quantitative Aptitude Questions | CAT Number Systems - Factors

The question is about number of factors of a number. Number of factors of abc is given. We need to find out the factors of abcabc. Dealing with factors of a number is a vital component in CAT Number Systems: Factors. A range of CAT questions can be asked based on this simple concept of Factors. CAT exam has always tested the idea of Factors from Number systems. The idea of Factors questions forms an integral part of the CAT syllabus. Make use of 2IIMs Free CAT Questions, provided with detailed solutions and Video explanations to obtain a wonderful CAT score.

Question 12: If a three digit number ‘abc’ has 2 factors (where a, b, c are digits), how many factors does the 6-digit number ‘abcabc’ have?

1. 16
2. 24
3. 18
4. 30

Explanatory Answer

Method of solving this CAT Question from Number Theory - Factors: Beauty of the number 1001. This number is not prime, is a product of three distinct primes and does wonderful things to three-digit numbers when multiplied to them.

To start with ‘abcabc’ = ‘abc’ * 1001 or abc * 7 * 11 * 13 (This is a critical idea to remember).
‘abc’ has only two factors. Or, ‘abc’ has to be prime. Only a prime number can have exactly two factors. (This is in fact the definition of a prime number)

So, ‘abcabc’ is a number like 101101 or 103103.
’abcabc’ can be broken as ‘abc’ * 7 * 11 * 13. Or, a p * 7 * 11 * 13 where p is a prime.
As we have already seen, 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, p * 7 * 11 * 13 will have = (1 + 1) * (1 + 1) * (1 + 1) * (1 + 1) = 16 factors

The question is "How many factors does the 6-digit number ‘abcabc’ have?"

Hence the answer is 16 factors.

Choice A is the correct answer.