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Explore the Back-StoryIf three prime numbers, all greater than , are in A.P. , then their common difference

(A) must be divisible by but not necessarily by ;

(B) must be divisible by but not necessarily by ;

(C) must be divisible by both and ;

(D) need not be divisible by any of and ;

**Discussion:**

Say and are the three primes in A.P. with d as the common difference.

Since , hence is odd.

If is odd, then is even. But then cannot be a prime any more. Hence contradiction. Thus cannot be odd. Hence divides .

Again implies is not divisible by . Hence p is either or modulo . Now we wish to check if d is divisible by or not. Suppose it is not then d is also either or modulo .

**Case 1:** is mod and is mod , then is mod hence contradiction (as p+2d is a prime greater than 3).

**Case 2:** is mod and is mod , then is mod hence contradiction (as p+d is a prime greater than 3)

**Case 1:** is mod and is mod , then is mod hence contradiction (as p+d is a prime greater than 3).

**Case 2:** is mod and is mod , then is mod hence contradiction (as p+2d is a prime greater than 3)

Hence must be modulo .

Therefore common difference must be divisible by both and .

**Answer: (C)**

If three prime numbers, all greater than , are in A.P. , then their common difference

(A) must be divisible by but not necessarily by ;

(B) must be divisible by but not necessarily by ;

(C) must be divisible by both and ;

(D) need not be divisible by any of and ;

**Discussion:**

Say and are the three primes in A.P. with d as the common difference.

Since , hence is odd.

If is odd, then is even. But then cannot be a prime any more. Hence contradiction. Thus cannot be odd. Hence divides .

Again implies is not divisible by . Hence p is either or modulo . Now we wish to check if d is divisible by or not. Suppose it is not then d is also either or modulo .

**Case 1:** is mod and is mod , then is mod hence contradiction (as p+2d is a prime greater than 3).

**Case 2:** is mod and is mod , then is mod hence contradiction (as p+d is a prime greater than 3)

**Case 1:** is mod and is mod , then is mod hence contradiction (as p+d is a prime greater than 3).

**Case 2:** is mod and is mod , then is mod hence contradiction (as p+2d is a prime greater than 3)

Hence must be modulo .

Therefore common difference must be divisible by both and .

**Answer: (C)**

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