Question

What is the product of the greatest prime number that is less than 50 and the smallest prime number that is greater than 50?
$
  {\text{A}}{\text{. 2491}} \\
  {\text{B}}{\text{. 2487}} \\
  {\text{C}}{\text{. 2489}} \\
  {\text{D}}{\text{. 2493}} \\
$

Answer 1

Hint – To find the product, we consider the formula of general form of a prime number and find out the nearest prime number below 50 and the nearest prime number above 50. We multiply these two numbers to determine the answer.

Complete step-by-step answer:
A prime number is defined as a number which is only divisible by 1 and itself.
The general form of a prime number is given by the formula 6n + 1 or 6n – 1, n can take all values except the multiples of prime numbers. (i.e. 2, 3, 5, 7, 11).
The greatest prime number that is less than 50 is:
6n + 1 < 50 or 6n – 1 < 50
⟹6n < 49 or 6n < 51
⟹n < 8 or n < 8
Substitute n = 8 in both 6n + 1 and 6n -1
⟹6(8) + 1 or 6(8) – 1
⟹49 or 47.
49 is not a prime because it is a multiple of 7.
Therefore the greatest prime number below 50 is “47” (It is only divisible by 1 and itself).

The smallest prime number that is greater than 50 is:
6n + 1 > 50 or 6n – 1 > 50
⟹6n > 49 or 6n > 51
⟹n > 8 or n > 8
Substitute n > 8 in both 6n + 1 and 6n -1
Let us take n = 9
⟹6(9) + 1 or 6(9) – 1
⟹55 or 53.
55 is not a prime because it is a multiple of 11and 5.
Therefore the smallest prime number above 50 is “53” (It is only divisible by 1 and itself).
Therefore the product of the greatest prime number that is less than 50 (i.e. 47) and the smallest prime number that is greater than 50 (i.e. 53) is
47 × 53 = 2491.
Option A is the correct answer.

Note – In order to solve this type of question the key is to know the definition of a prime number and its general form.
We can also solve this problem by trial and error, we could randomly select a number above and below 50, check if it’s prime or not by writing down its divisors and determine the greatest prime below 50 and least prime above 50 and find their product.