Question

Show that every even positive integer is in the form 5m+1 or 5m+3 for some odd integer m

Answer 1

Let N be any positive integer.


On dividing N by 5 , Let M be Quotient and R be Remainder.


Then , By Euclid division lemma , we have

N = 5M+1 , where r = 0,1,2,3,4

N = 5M , where R = 0

N =5M + 1 , where R = 1

N = 5M + 2 , where R = 2

N = 5M + 3 , where R = 3

N = 5M + 4 , where R = 4

N = 5M , 5M+2 , 5M+4 are even values of N.

Thus

When N is odd , it is in the form of 5M , 5M+1 and 5M+3.