Question

Prove that any positive odd integer is of the form of 5q+1,5q+3,for some integer is q

Answer 1

$\large{\underline{\boxed{\tt Answer}}}$

Proved!!

$\large{\underline{\boxed{\tt Step-by-step \ Explanation}}}$

Given Probelm:

Prove that any positive odd integer is of the form of 5q+1,5q+3,for some integer is q

To Do:

Prove

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Method:

⇒Any integer can be written in the form 5m , 5m+1, 5m+2 (where m is any integer)

⇒(5m)² = 25m² = 5 X 5m² = 5q (where q = 5m²)

⇒(5m+1)² = (5m)² + 2X 5m X 1 + 1²

By using identity- (a+b)²= a²+ 2ab +b²

⇒25m² + 10 m + 1

⇒5 (5m²+ 2m) +1

⇒ 5q +1 ( q= 5m²+2m)

(5m+2)²= (5m)² + 2X 5m X 2 +2²

By using identity---(a+b)²= a²+ 2ab +b²

⇒ 25m² + 20 m +4

⇒ 5 (5m²+4m ) + 4

⇒ 5q+4 (where q= 5m²+4m)

Hence,

$\huge\boxed{Its \ Proved!}$

Answer 2

Answer:

let a be an odd positive integer and b=5 where q is any even integer

then,

a = bq +r

a = 5q + r

then,

0≤r<5

r= 0,1,2,3,4,

now,

when r=0

then,

a= 5q

when r= 1

then,

a= 5q +1

when r=2

then,

a= 5q+ 2

when r=3

then,

a= 5q+3

when r=4

then,

a= 5q+ 4

since, 5q , 5q +2 , 5q +4 are even integers as q is an even integer.

So,

a≠ 5q , 5q +2 , 5q +4

Hence, any odd positive integer is the form of 5q+1, and 5q+3

Hope this helps you.....