himu wrote:Q. If p and q are two different odd prime numbers such that p < q, then which of the following must be true?
A. (2p + q) is a prime number
B. p + q is divisible by 4
C. q - p is divisible by 4
D. (p + q + 1) is the difference between two perfect squares of integers is the difference between two perfect squares of integers
E. (p^2+q^2) is the difference between two perfect squares of integers
Try to prove that the answer choices DON'T have to be true.
A: 2p+q is prime
If p=5 and q=11, then 2p+q = 2*5 + 11 = 21, which is NOT prime.
Eliminate A.
B. p + q is divisible by 4
If p=5 and q=13, then p+q = 5+13 = 18, which is NOT divisible by 4.
Eliminate B.
C. q - p is divisible by 4
If p=5 and q=19, then q-p = 19-5 = 14, which is NOT divisible by 4.
Eliminate C.
E: (p²+q²) is the difference between two perfect squares of integers
The difference between two perfect squares = a² - b².
If p=3 and q=5, then p²+q² = 3²+5² = 34.
Implication:
a² - b² = 34
(a+b)(a-b) = 34.
.
Case 1: (a+b)(a-b) = 34*1.
Adding together a+b = 34 and a-b = 1, we get:
(a+b) + (a-b) = 34+1
2a = 35.
a = 35/2.
Not viable, since a must be an integer.
Case 2: (a+b)(a-b) = 17*2.
Adding together a+b = 17 and a-b = 2, we get:
(a+b) + (a-b) = 17+2
2a = 19.
a = 19/2.
Not viable, since a must be an integer.
Thus, p²+q² = 34 cannot be equal to the difference of two perfect squares.
Eliminate E.
The correct answer is
D.
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