Show that any positive odd integer is of the form 6q+1, or 69 +3, or 69 + 5, where q some integer. Question Show that any positive odd integer is of the form 6q+1, or 69 +3, or 69 + 5, where qsome integer. in progress 0 Math Audrey 1 month 2021-08-20T19:57:58+00:00 2021-08-20T19:57:58+00:00 2 Answers 0 views 0

## Answers ( )

if ‘a’ and ‘b’ are two positive integers then

where,

0 ≤ r < blet a be a positive integers and b = 6

where,

0 ≤ r < 6

from above we can say that remainder is less than 6 and equal or greater than 0

so the possible values of r be

by putting values of divisor(b) and remainder in equation

putting r = 0, a = 6q + 0

a = 6q

it is divided by 2 so it is an number.

putting r = 1, a = 6q + 1

6q is divided by 2 but 1 is not divided.

it is not divided by 2 so it is an number.

putting r = 2, a = 6q + 2

it is divided by 2 so it is an number.

putting r = 3, a = 6q + 3

it is not divided by 2 so it is an number.

putting r = 4, a = 6q + 4

it is divided by 2 so it is an number

putting r = 5, a = 6q + 5

it is not divided by 2 so it is an number.

so by these we can say that any positive odd integer is of the form 6q+1, or 69 +3, or 69 + 5, where q is some integer.

lets us start with taking a,where a is a postive off integerWe apply the division algorithm with a and b=6since 0 less than or equal to r or lessthan 6 the possible remainders are 012345that isa can be6q+1or6q+3or6q+5where quotienthowever since a is odd a cannot be 6q,6q+2,6q+4since they are divisble by 2therefore the any odd integer is of the form of 6q+1,6q+3,6q+5Thank You