Q. If p and q are two different odd prime numbers such that p < q, then which of the following must be true?
1.(2p + q) is a prime number
2. p + q is divisible by 4
3. q – p is divisible by 4
4. (p + q + 1) is the difference between two perfect squares of integers.
5. \((p^2+q^2)\) is the difference between two perfect squares of integers[/quote]
Analysis of the options :
1. (2p + q) for sure is an odd number , but may or may not be a prime number .
Eg - if we take 3 and 5 as p and q resp, we get 11, which is a prime number.
But if we take 7 and 11, we get 25, which is not a prime number.
2. p + q for sure gives us an even number. It may or may not be divisible by 4.
Eg - (7 + 5) = 12 , which is divisible by 4. But (11 + 7) = 18 , not divisible by 4
3. Same way as 2nd.
4. (p + q + 1) gives us an odd number.
Lets see the trend of difference between squares of consecutive integers.
2^2 - 1^2 = 3
3^2 - 2^2 = 5
4^2 - 3^2 = 7
5^2 - 4^2 = 9
6^2 - 5^2 = 11 and so on.
The trend is : The difference between squares of consc. integers is always odd number. And it covers all the odd numbers.
However , when we don't take consecutive integers, this trend is not followed.
Eg : 5^2 - 3^2 = 16 , which is an even number
3^2 - 1^2 = 8 , again not an odd number
Hence : Any odd number can be expressed as the difference of the squares of integers.
5. (p^2 + q^2) gives an even number.
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