Please solve question 3,5

Dear Student,

3).

Any odd positive integer is of the form 4m + 1 or 4m + 3 for some integer m.

When n = 4m + 1,

n^{2} - 1 = {(4 m + 1)}^{2} - 1 = 16 m^{2} + 8 m + 1 - 1 = 16 m^{2} + 8 m = 8 m (2 m + 1) \Rightarrow n^{2} - 1 i s d i v i s i b l e b y 8 W h e n n = 4 m + 3 n^{2} - 1 = {(4 m + 3)}^{2} - 1 = 16 m^{2} + 24 m + 9 - 1 = 16 m^{2} + 24 m + 8 = 8 m (2 m^{2} + 3 m + 1) \Rightarrow n^{2} - 1 i s d i v i s i b l e b y 8

Hence, n² – 1 is divisible by 8 if n is an odd positive integer.

For remaining queries, we request you to post them in separate threads to have rapid assistance from our experts.

Regards!

View Full Answer