Show that any positive odd integers is of the form 4q + 1 or 4q+3 . Where q is some integers
Answers
Answer :
Let "a" be the positive odd integer which when divided by 4 gives q as quotient and "r" as remainder.
According to Euclid's division lemma :
a = bq + r, where 0 ≤ r < b
Here b = 4 and 0 ≤ r < 4
a = 4q + r
=> a = 4q + 0 (when r = 0)
Even
=> a = 4q + 1 (when r = 1)
Odd
=> a = 4q + 2 (when r = 2)
Even
=> a = 4q + 3 (when r = 3)
Odd
So, a = 4q+1 or, a = 4q+3 are odd
Hence, any positive odd integer is of the form 4q+1, 4q+3
To Show:
Any positive odd integers is of the form 4q + 1 or 4q+3 . Where q is some integers
Solution:
Let a be any positive integer.
By Euclid's Division Lemma,
a = bq + r
So, the possible remainders are 0, 1, 2 and 3.
=> a can be of the form 4q, 4q+1, 4q+2 and 4q+3.
where q is the quotient .
a is odd and hence cannot be of the form 4q or 4q + 2 as they are even.
So, a will be in the form of 4q + 1 or
4q + 3.
Hence proved.