Q25. Consider the following in respect of prime number p and composite number c.
1. 𝑝+𝑐 / 𝑝−𝑐 can be even.
2. 2p+c can be odd.
3. pc can be odd.
Which of the statements given above are correct?
Detailed Solution
Let's evaluate each statement: 1. (p+c)/(p-c) can be even. Consider p = 11 (an odd prime) and c = 9 (an odd composite number). p+c = 11+9 = 20 (even) p-c = 11-9 = 2 (even) (p+c)/(p-c) = 20/2 = 10, which is an even number.
So, statement 1 is correct. 2. 2p+c can be odd. Since p is an integer, 2p will always be an even number. For 2p+c to be odd, c must be an odd number (Even + Odd = Odd). Consider p = 3 (an odd prime) and c = 9 (an odd composite number).
2p+c = 2(3) + 9 = 6 + 9 = 15, which is an odd number. So, statement 2 is correct. 3. pc can be odd. For the product of two integers to be odd, both integers must be odd. Consider p = 3 (an odd prime) and c = 9 (an odd composite number).
pc = 3 × 9 = 27, which is an odd number. So, statement 3 is correct. Since all three statements can be true, the correct option is D.
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