CAT Practice : Number System: Factors

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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.

Number Theory 'abcabc'

    Q.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


  • Correct Answer
    Choice C. 16 or 24 factors.

Detailed Solution

‘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 paqbrc 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.
Choice (C)

Correct Answer: 16 or 24 factors.

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