Quant Question Of The Day: 101
Numbers
Which of the following statements are always correct :-
I. If p and p² + 8 are prime numbers, then p³ + 8p + 2 is also prime.
II. a4 + 4 is prime for some a > 1.
a) Only I
b) Only II
c) Both
d) None
I. p and p² + 8 both are prime
Since p is prime, we can say that for p ≥ 5, p = 6k ± 1
=> p² + 8 = 36k² ± 12k + 9
(a multiple of 3, so can not be prime)
=> p can only be 3
For p = 3, p³ + 8p + 2 = 53 which is prime
So, correct
II. a4 + 4 = (a² + 2)² – 4a² = (a² + 2)² = (2a)²
=> a4 + 4 = (a² – 2a + 2)(a² + 2a + 2)
Now, (a² + 2a + 2) = (a + 1)² + 1, so always greater than 5 for a > 1
(a² – 2a + 2) = (a – 1)² + 1, so always greater than or equal to 2 for a > 1
Hence always composite for a > 1
See our previous ‘Questions of the Day’:
Quant Question Of The Day: 100
Quant Question Of The Day: 99
d
Ans:-a
Answer is a