Question

sum of how many pairs of prime numbers give a total of 100 ​

Answer 1

QUESTION :-

sum of how many pairs of prime numbers give a total of 100 ?

SOLUTION :-

All primes (except 2 ) are odd. This means that adding any two primes (except when one of them is 2 ) will result in an even number which cannot be prime. So, if a pair of primes adds up to be a prime, then one of them must be 2 . Consequently, the task at our hands is to find all values of n (between 1 and 100 ), such that n−1 and n+1 are primes.

Let's first identify the first two trivial solution pairs:-

2,3 (2+3=5)

2,5 (2+5=7)

and get them out of our way.

Now for the remaining solutions, let's start putting conditions on n, so that we can reduce our effort using elimination:

1. Neither n−1 nor n+1 is divisible by 2 . This means that n is divisible by 2 .
2. Neither n−1 nor n+1 is divisible by 3 . This means that n is divisible by 3 . The first two conditions put together mean that n is divisible by 6 .
3. Neither n−1 nor n+1 is divisible by 5 . So the last digits of n−1 and n+1 are neither 0 nor 5 . This means that the last digits of n cannot be 1,4,6 or 9 . The first three conditions put together mean that n could be any multiple 6 that does not end with 4 or 6 . This limits our possibilities of n to only 9 numbers: 12,18,30,42,48,60,72,78,90
4. Neither n−1 nor n+1 is divisible by 7 . We have already accounted for all the primes that are smaller than 7 . So now we only need to eliminate the factor combinations involving 7 and higher primes. With this we can eliminate three more values of n in which either n−1 or n+1 is divisible by 7 . They are: 48 (because 7×7=49 ), 78 (because 7×11=77 ) and 90 (because 7×13=91 ). All the other factor combinations of 7 and higher primes will result in numbers bigger than 100 that we are not interested in.

With this, we now have 6 possible values of n left : 12,18,30,42,60,72 .

Add the first two trivial cases to this and there you go. We have eight possible solutions.

2,3

2,5

2,11

2,17

2,29

2,41

2,59

2,71

2,3 2,5 2,11 2,17 2,29 2,41 2,59 2,71

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