show that any positive odd integers is of the form 6q+1 or 6q+3 or 6q+5 where q is some integers
Answers
Answer:
Let 'a' be any positive integer and b=6
Apply Euclid division lemma to A and B
r=0,1,2,3,4,5
a=6q,6q+1,6q+2,6q+3,6q+4,6q+5
and. a is positive odd integer
a≠6q. or a≠ 6q+2 or a≠6q+4
And. a=6q+1 ,a=6q+3 , a=6q+5
Hence proved
Let ‘a’ be any positive integer.
Then from Euclid's division lemma,
a = bq+r ; where 0 < r < b
Putting b=6 we get,
⇒ a = 6q + r, 0 < r < 6
For r = 0, we get a = 6q = 2(3q) = 2m, which is an even number. [m = 3q]
For r = 1,
We get a = 6q + 1 = 2(3q) + 1 = 2m + 1, which is an odd number. [m = 3q]
For r = 2,
We get a = 6q + 2 = 2(3q + 1) = 2m, which is an even number. [m = 3q + 1]
For r = 3,
We get a = 6q + 3 = 2(3q + 1) + 1 = 2m + 1, which is an odd number. [m = 3q + 1]
For r = 4,
We get a = 6q + 4 = 2(3q + 2) + 1 = 2m + 1, which is an even number. [m = 3q + 2]
For r = 5,
We get a = 6q + 5 = 2(3q + 2) + 1 = 2m + 1, which is an odd number. [m = 3q + 2]
Thus, from the above it can be seen that any positive odd integer can be of the form 6q +1 or 6q + 3 or 6q + 5, where q is some integer.