Question

If a 3 digit number 'abc' has 3 factors, how many factors does the 6-digit number 'abcabc' have?

• A. 16 factors
• B. 24 factors
• C. 16 or 24 factors
• D. 20 factors

Answer 1

The correct option is C

16 or 24 factors

'abc' has exactly 3 factors, so 'abc' should be square of prime number.
Let the square root of 'abc' be 'p' (where 'p' is prime).

'abcabc' = abc $\times$ 1001 = $p^2\times 13\times 11\times 7$

If 'p' is any prime number other than 13, 11 or 7, then no. of factors = 3 x 2 x 2 x 2 = 24 (no. will be of the form $p^2\times 13\times 11\times 7$)

If 'p' is any prime number among 13, 11 or 7, then no. of factors = 4 x 2 x 2 = 16 (no. will be of the form $7^3\times 13\times 11,ifp=7$)

$\therefore$ No. of factors can be 16 or 24.