prime numbers explanation

Question

Can someone pls explain me the significance of : If is a positive integer greater than 1, then there is always a prime number with n<n<2n

also, all prime numbers above 3 are of the form 6n-1 or 6n+1 , because all other numbers are divisible by 2 or 3. examples to both will be of great help

regards

KarishmaB · Answer

This is the Bertrand's postulate: For every n > 1 there is always at least one prime p such that n < p < 2n. 
The proof is complicated. It has no significance as far as GMAT is concerned. You are not expected to know Bertrand's postulate and this is not an intuitive number property.

The second property you mentioned is important. This is how you can explain it.

Every integer equal to or greater than 6 will be of one of the 6 forms: 6n or 6n+1 or 6n+2 or 6n+3 or 6n+4 or 6n+5

Note here that 6n is divisible by 6.
6n+2 = 2(3n+1) i.e. divisible by 2
6n+3 = 3(2n+1) i.e. divisible by 3
6n+4 = 2(3n+2) i.e. divisible by 2

Hence none of these can be prime. So a prime must be of the form 6n+1 or 6n+5 (which is the same as 6n-1)

Notice that 5 can be written as 6n - 1 where n = 1.
Only 2 and 3 cannot be written in one of the two forms: 6n+1 or 6n - 1.

Hence every prime number greater than 3 will be of one of these forms. (mind you, it does NOT mean that every number of the form 6n+1 or 6n-1 will be prime)

sdas · Answer

Awesome. that helps. Thank you Karishma

sdas · Answer

If a is a factor of bc, and gcd(a,b)=1 then a is a factor of c

sdas · Answer

OK. got this one. also, if gcd(a,b) is 1 then both numbers are relatively prime

prime numbers explanation

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards