badge image

Enroll For Free Now & Improve Your Performance.

×
User Icon
User Icon
User Icon
User Icon
User Icon

Thank you for registering.

One of our academic counsellors will contact you within 1 working day.

Please check your email for login details.
MY CART (5)

Use Coupon: CART20 and get 20% off on all online Study Material

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: Rs.

There are no items in this cart.
Continue Shopping
Menu
Grade: 12

                        

. Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

. Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

3 months ago

Answers : (1)

Anand Kumar Pandey
askIITians Faculty
2937 Points
							Dear Student

Let x be any positive integer and y = 3.
By Euclid’s division algorithm, then,

x = 3q + r for some integer q≥0 and r = 0, 1, 2, as r ≥ 0 and r < 3.
Therefore,
x = 3q, 3q+1 and 3q+2

Now as per the question given, by squaring both the sides,
we get,
x^2=(3q)^2
=9q^2
=3×3q^2
Let 3q^2= m
Therefore
x^2= 3m ..........................(1)

x^2= (3q + 1)^2
= (3q)^2+1^2+2×3q×1
= 9q^2+ 1 +6q
= 3(3q^2+2q) +1
Substitute,
3q2+2q = m, to get,
x^2= 3m + 1 .................................. (2)


x^2=(3q+2)^2
=(3q)^2+2^2+2×3q×2
=9q^2+4+12q
=3(3q^2+4q+1)+1
Again, substitute, 3q^2+4q+1 = m, to get,

x^2= 3m + 1................................. (3)

Hence, from equation 1, 2 and 3, we can say that, the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

Thanks
3 months ago
Think You Can Provide A Better Answer ?
Answer & Earn Cool Goodies


Course Features

  • 728 Video Lectures
  • Revision Notes
  • Previous Year Papers
  • Mind Map
  • Study Planner
  • NCERT Solutions
  • Discussion Forum
  • Test paper with Video Solution


Course Features

  • 731 Video Lectures
  • Revision Notes
  • Test paper with Video Solution
  • Mind Map
  • Study Planner
  • NCERT Solutions
  • Discussion Forum
  • Previous Year Exam Questions


Ask Experts

Have any Question? Ask Experts

Post Question

 
 
Answer ‘n’ Earn
Attractive Gift
Vouchers
To Win!!! Click Here for details