‘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.
Choice (C)
Correct Answer: 16 or 24 factors.
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