akhil911 wrote:

x is the smallest integer greater than 1000 that is not prime and that has only one factor in common with 30!. What is x?

A. 1009

B. 1021

C. 1147

D. 1273

E. 50! + 1

Good question found this on

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Certainly one of our tricky questions with a great takeaway!

If we start with the options, the way we often do when dealing with prime/composite numbers, we would get stuck. We are looking for a non-prime number so quite probably some of the options would be prime. Much as we might try, we will not find a factor for them and hence waste a lot of precious time.

Here, we do need to start from the question stem only.

We need an integer that has nothing other than 1 in common with 30!. Which is the smallest such integer? It is 31, the next prime number.

Similarly, if you were looking for the smallest number with no factor other than 1 in common with 31!, you would skip to 37. All integers between 31 and 37 are composite and hence would have factors lying between 1 and 31. Note that every composite number is made of prime factors. All composites less than 37 will be made up of 2 or more prime numbers less than 37.

32 = 2*2*2*2*2

33 = 3*11

34 = 2*17

35 = 5*7

36 = 2*2*3*3

This concept is extremely important.

If you want to make a composite number which has no primes till 30, you must use two or more primes greater than 30. The smallest prime greater than 30 is 31.

If you use two of them, you get 31*31 = 961 (not more than 1000)

So the two primes should be 31*37 = 1147 (greater than 1000)

Answer (C)

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