CAT 1991 — QA Question 46
Answer the following questions based on the information given below:
There were a hundred schools in a town. Of these, the number of schools having a play – ground was 30, and these schools had neither a library nor a laboratory. The number of schools having a laboratory alone was twice the number of those having a library only. The number of schools having a laboratory as well as a library was one fourth the number of those having a laboratory alone. The number of schools having either a laboratory or a library or both was 35.
A positive integer is said to be a prime number if it is not divisible by any positive integer other than itself and 1. Let p be a prime number greater than 5. Then (p2 – 1) is
Answer & solution
- A
never divisible by 6
- B
always divisible by 6, and may or may not be divisible by 12
- C
always divisible by 12, and may or may not be divisible by 24
always divisible by 24
As each prime number greater than or equal 3 is of from 6k ± 1, here k is a natural number, p = 6k ± 1.
p2 - 1 = (6k ± 1)2 - 1 = 12k(3k ± 1)
Since k(3k ± 1) is an even number, above number is always divisible by 24 irrespective of value of k.
Hence, option (d).