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.CAT exam has been testing these type of questions for the past few decades and 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 8: If a three digit number ‘abc’ has 3 factors, how many factors does the 6-digit number ‘abcabc’ have?

1. 16 factors
2. 24 factors
3. 16 or 24 factors
4. 20 factors

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.

‘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.

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

Hence the answer is 16 factors or 24 factors.

Choice C is the correct answer.