Consider all prime number pairs separated by just one number (like <5,7>, <11,13>….etc). We can clearly observe that the number in between is divisible is 6.
Can we prove that this is always be true for any pair of prime numbers.
Solution :
Let the prime numbers be n, n+2. Our case is to prove n+1 is divisible by 6, ie n+1 is divisible by 2 and 3.
In this case – since n is prime, it will be odd – so we can easily say n+1 will be even, divisible by 2
Next is to prove it is divisible by 3 as well. This proved to be baffling for me.
Later realized – for any 3 consecutive numbers, atleast one of them should always be divisible by 3. This is true for any 3 consecutive numbers as multiples of 3 repeat at that interval.
So definitely one of n,n+1,n+2 should be divisible by 3. Since n and n+2 are prime – they cannot be multiples of 3 – so n+1 should be a multiple of 3.