CAT Practice : Number System: Factors

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Given prime factorization, getting to the number of factors is easy. Given number of factors, how does one get to the prime factorization? Chew on that.

Number Theory - 18 factors

    Q.10: Find the smallest number that has exactly 18 factors.
    1. 180
    2. 216
    3. 240
    4. None of these

 

  • Correct Answer
    Choice A. 180.

Detailed Solution

Any number of the form paqbrc will have (a + 1) (b + 1) (c + 1) factors, where p, q, r are prime. (This is a very important idea)
Now, the number we are looking for has 18 factors. It can comprise one prime, two primes or three primes.

Now, 18 can be written as 1 * 18 or 3 * 6 or 9 * 2 or 2 * 3 * 3.

If we take the underlying prime factorization of N to be paqb, then it can be of the form
p1q8 or p2q5

If we take the underlying prime factorization of N to be pa, then it can be of the form
p17

If we take the underlying prime factorization of N to be paqbrc, then it can be of the form
p1q1r2
So, N can be of the form p17, p2q5, p1q8 or p1q2r2

Importantly, these are the only possible prime factorizations that can result in a number having 18 factors.

Now, let us think of the smallest possible number in each scenario
p17 - Smallest number = 217
p2q5 – 32 * 25
p1q8 – 31 * 28
p1q2r2 – 51 * 32 * 22

The smallest of these numbers is 51 * 32 * 22 = 180

Correct Answer: 180



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