Show that every positive odd integer is of the form 4q + 1 or 4q+3, where q is
some integer
Answers
Answer:
from the following plants which the them have flowers did you find any flower which has no difference between stress and petals did you find any flower which the number of system is different from the number of plate Petals.
Answer:
Given :-
A positive integer
To find :-
Show that "Every positive odd integer is of the form 4q +1 and 4q+3,where q is some integer".
Solution :-
We know that
Euclid's Division Lemma:
For any two positive integers a and b there exist two positive integers q and r satisfying a = bq+r, 0≤r<b.
Let consider a = 4q+r ------------(1)
The possible values of r = 0,1,2,3
I) If r = 0 then
a = 4q+0
=>a = 4q
=> a = 2(2q)
=> a = 2m -----------(2)
Where m = 2q
ii) If r = 1 then
a = 4q+1-----------(3)
iii) If r = 2 then
a = 4q+2
=> a = 2(2q+1)
=> a = 2m ----------(4)
Where m = 2q+1
iv) If r = 3 then
a = 4q+3 -----------(5)
From (2)&(4)
a is the positive even number.
From (3)&(5)
a is the positive odd number.
Every positive odd integer is of the form 4q +1 and 4q+3,where q is some integer.
Hence, Proved.
Used formulae:-
Euclid's Division Lemma:-
For any two positive integers a and b there exist two positive integers q and r satisfying a = bq+r, 0≤r<b.
Step-by-step explanation: