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Question: Show that any positive odd integer is of the form \(4q + 1\) or \(4q + 3\) where q is the positive i...

Show that any positive odd integer is of the form 4q+14q + 1 or 4q+34q + 3 where q is the positive integer.

Explanation

Solution

we can use the Euclid division lemma which states that, If a and b are two positive integers then we can write,
a=bq + r where 0⩽r<b0 \leqslant r < b
Now assume the positive integers to be a and b=44 and put it in the formula. Then put the value of r to find if the equation gives an odd positive integer or not.

Complete step by step solution:
We have to show that any positive odd integer is of form 4q+14q + 1 or 4q+34q + 3 where q is the positive integer.
On using Euclid’s division lemma, If a and b are two positive integers then we can write,
a=bq + r where 0⩽r<b0 \leqslant r < b
Then let the positive integers be a and b=44
Then we can write,
a=4q+r4q + r--- (i) where 0⩽r<40 \leqslant r < 4
There here r can be either greater or equal to zero and less than 44
So if r=11 then eq. (i) becomes-
⇒a=4q+1\Rightarrow a = 4q + 1
Now this will always be an odd integer whether we put any positive integer value for q.
Now if we put r=33 then eq. (i) becomes-
⇒a=4q+3\Rightarrow a = 4q + 3
Now this will always be an odd integer whether we put any positive integer value for q.

Hence Any positive odd integer is of form 4q+14q + 1 or 4q+34q + 3 where q is the positive integer, Proved.

Note:
Here we have not put r=22 because then we will get,
⇒a=4q+2\Rightarrow a = 4q + 2
And if we put any positive integer value in place of q we will get only positive even numbers.