Question

Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer​

Answer 1

Let b=6

a=bq+r

a=6q+r

Possibility of r of odd integer =1,3,5

When r=1

6q+1=a

When r=3

6q+3=a

When r=5

6q+5=a

............I hope this will help you........

Answer 2

From euclids lemma,
a=bq+r
(where a and b are integers and 0=/<r < b)
Let a be any positive integer
Let b=6
Therefore, r=0,1,2,3,4or5
Case 1(when r=0)
a=6q+0
(here a will be even since 6 x q results in even integer)
Case 2(r=1)
a=6q+1
(here a will be of since 6 x q is even and even +1 results in our integer)
Case 3(r=2)
a=6q+2
(here a will be even since 6 x q is even and even +2 results in even integer)
Case 4(r=3)
a=6q+3
(here a will be odd since 6 x q is even and even +3 results in odd integer)
Case 5(r=4)
a=6q+4
(here a will be even since 6 x q is even and even +4 results in even integer)
Case 6(r=5)
a=6q+5
(here a will be odd since 6 x q is even and even + 5 results in odd integer)
Since we got a as an odd integer in cases 2, 4and 6 only,
Therefore, it can be stated that and positive odd integer is of the 6q+1 or 6q+3 or 6q+5
Hence Proved


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