Explanation:

‘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.