Question

show that any positive odd integer is of the form of (4q + 1) or (4q +3) where q is some integer ..​

Answer 1

Step-by-step explanation:

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Answer 2

$\huge{\underline{\bf{\blue{Question:-}}}}$

show that any positive odd integer is of the form of (4q + 1) or (4q +3) where q is some integer ..

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$\large{\underline{\bf{\pink{Answer:-}}}}$

$\large{\underline{\bf{\purple{Explanation:-}}}}$

$\large{\underline{\bf{\green{To\:Find:-}}}}$

we need to show that any positive odd integer is of the form of (4q + 1) or (4q +3) where q is some integer.

$\huge{\underline{\bf{\red{Solution:-}}}}$

Let n be any odd positive intiger.

On dividing n by 4 ,

let q be the quotient and r be the remainder.

So,

By Euclid Division lemma, we have

⠀⠀⠀⠀⠀n = 4q + r , where 0 ≤ r < 4

∴ n = 4q or ( 4q + 1 ) or ( 4q + 2 ) or ( 4q + 3 )

Clearly , 4q or (4q + 2 ) are even and since n is odd ,

• So , n ≠ 4q
• and n ≠ ( 4q + 2 ).

∴ n = ( 4q + 1 ) or (4q + 3) , for some intiger q.

Hence , any positive intiger odd intiger is of the form ( 4q + 1) or (4q + 3 )for some intiger q

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