Question

Show that every even positive integer is of the form 5m+1 5m+3 for some integer.

Answer 1

Heya !!!

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.

HOPE IT WILL HELP YOU.... :-)

Answer 2

hey

here is answer

let a be any positive integer

then

b=5

0≤r<b

0≤r<5

r=0,1,2, 3,4

case 1.

r=0

a=bq+r

5q+0

5q

case 2.
r=1
a=bq+r

5q+1

case 3.

r=2

5q+2

case4.

r=3
5q+3

case 5.

r=4

5q+4


note= i have taken q instead of m
from above it is proved.

hope it helps

thanks