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

Answers (1)

Our aim is to prove that the square of any positive integer is of the form 5q or,5q+1 or,5q+4 for some integer q.

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.

\begin{align*} 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\\ \end{align*}

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

Preparation Products

Knockout JEE Main April 2021 (One Month)

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Weekend Live Classes, Mentorship from our Experts, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 14000/- ₹ 4999/-
Buy Now
Knockout NEET Aug 2021 (One Month)

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Weekend Live Classes, Mentorship from our Experts, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 14000/- ₹ 4999/-
Buy Now
Knockout JEE Main May 2021

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Weekend Live Classes, Mentorship from our Experts, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 22999/- ₹ 9999/-
Buy Now
Knockout NEET Aug 2021

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Weekend Live Classes, Mentorship from our Experts, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 22999/- ₹ 9999/-
Buy Now
Test Series JEE Main May 2021

Unlimited Chapter Wise Tests, Unlimited Subject Wise Tests, Unlimited Full Mock Tests, Get Personalized Performance Analysis Report,.

₹ 6999/- ₹ 2999/-
Buy Now
Boost your Preparation for JEE Main with our Foundation Course
 
Exams
Articles
Questions