CAT 2003 Slot 2 — QA Question 4
Prime and Composite NumbersEasy
If a, a + 2 and a + 4 are prime numbers, then the number of possible solutions for a is:
Answer & solution
one
- B
two
- C
three
- D
more than three
Solution
a, a + 2, a + 4 are prime numbers.
Now, all prime numbers greater than 3 are of the form 6k ± 1, where k is some natural number.
When a = 6k + 1, then a + 2 = (6k + 1) + 2 = 6k + 3 = 3(2k + 1), which is not prime since it is a multiple of 3
When a = 6k – 1, then a + 4 = (6k – 1) + 4 = 6k + 3 = 3(2k + 1), which is again not prime
So, the only possible values of a that remain are the prime numbers which are less than or equal to 3; i.e. 2 and 3.
When a = 2, a + 2 and a + 4 are obviously not prime numbers
When a = 3, a + 2 = 5 and a + 4 = 7, which satisfies the required condition
Hence, option (a).