Question

Express the prime number 31 in the form 2^p − 1 where p is a prime number and as a difference of two perfect squares using the identity (a +b)(a-b) = a²-b²

Answer 1

31 = 2⁵ -1

31 = 16²- 15²

Explanation:

As we know that

2¹=2

2²= 4

2³= 8

2⁴= 16

2⁵=32

So

32 = 2⁵

By subtracting 1 in bot sides

32-1 = 2⁵ -1

31 = 2⁵ -1

So

2⁵ -1 = 31

By comparing with

`2^p-1`

p= 5

As we know that

15²= 225

16² = 256

16²- 15² = 256-225 = 31

16²- 15² = (16+15)(16-15)

31 = (16+15)(16-15)