Question

Prove that square of any positive
integer is of the form 5q, 5q+1 ,5q+4
for some integer q .​

Answer 1

Answer:

here is the answer

Step-by-step explanation:

Lets first prove the first part, this is, lets prove that the square of any positive integer is of the form 5q or,5q+1 or 51 + 4.

Let us consider a an integer number such that a = 5m + r.

Applying rules of division algorithm we know that r = 0 or 0 < r < 5.

We need then to consider all cases of r. Thus:

r = 0

Lets also consider q to be equal to m^2.

When r = a we can conclude that a = 5m.

a = 5m

a^2 = ( 5m )^2

a^2 = 5 ( 5m^​2 )

a^2 = 5q.

r = 1

Lets also consider q to be equal to 5m^2 + 2m.

a = 5m + 1

a^2 = ( 5m + 1 )^2

a^2 = 25m^2 + 10m + 1

a^2 = 5 ( 5m^2 + 2m ) + 1

a^2 = 5q + 1.

r = 2

Lets also consider q to be equal to 5m^2 + 4m.

a = 5m + 2

a^2 = ​( 5m + 2 )^2

a^2 = 25m^2 + 20m + 4

a^2 = 5 ( 5m^2 + 4m ) + 4

a^2 = 5q + 4.

r = 3

Lets also consider q to be equal to 5m^2 + 6m + 1.

a = 5m + 3

a^2 = ​( 5m + 3 )^2

a^2 = 25m^2 + 9 + 30m

a^2 = 25m^2 + 30m ​ + 5 + 4

a^2 = 5 ( 5m^2 + 6m + 1 ) + 4

a^2 = 5q + 4.

r = 4

Lets also consider q to be equal to 5m^2 + 8m + 3.

a = 5m + 4

a^2 = (5m + 4)^2

a^2 = 25m^2 + 40m +15 + 1

a^2 = 5 ( 5m^2 + 8m + 3 ) + 1

a^2 = 5q + 1

Hence, the square of any positive integer is of the form 5q or 5q + 1 or 51 + 4.

hope this answer really helps you in solving this equation

Answer 2

let a be any positive integer..

Hope this will help you...