#### Thank you for registering.

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

Click to Chat

1800-5470-145

+91 7353221155

CART 0

• 0
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

# 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.

Latika Leekha
7 years ago
Let ‘a’ be a positive integer. now, we using the Euclid’s lemma we need to prove that the square of a is either of the form 3m, 3m+1 or 3m + 2.
Now, a = 3l + s, for some integer l $\geq$ 0, where s = 0, 1 or 2.
Therefore, a = 3l or 3l + 1 or 3l + 2.
Hence, a2 = (3l)2 or (3l + 1)2 or (3l + 2)2
= 9l2 , 9l2 + 6l + 1 or 9l2 + 12l + 4
= 3.3l2 or 3(3l2 + 2l) + 1 or 3(3l2 + 4l + 1) + 1
= 3.r1 or 3.r2 + 1 or 3.r3 + 1, where r1, r2, r3 are positive integers.
Hence, the square of a positive integer is either of the form 3m or 3m + 1.
Thanks & Regards
Latika Leekha