Question

Use exercise 37 to show that the integers $2^{35}-1,2^{34}-1,2^{33}-1,2^{31}-1,2^{29}-1$and$2^{23}-1$ are pair wise relatively prime.

Short answer

The given integers 35,34,33,31, 29 and 23 are pair wise relatively prime numbers.

Step-by-step solution

Step: 1

It is known that$\gcd \left(2^a-1,2^b-1\right)=2^{\gcd (a,b)}-1$

Step: 2

Check 35, 34, 33, 31, 29 and 23 are relatively prime or not.

$$\begin{gathered} 35=7.5 \\ 34=2.17 \\ 33=3.11 \\ 31=1.31 \\ 29=1.29 \\ 23=1.23 \end{gathered}$$

Step: 3

Therefore, there are no common factors of the numbers 35,34,33,31, 29 and 23.

So, they are relatively prime. It concludes that the given integers are pair wise relatively prime numbers.