Math, asked by jonasbvjose24, 4 months ago

Using Euclid's division lemma, show that the square of any positive integer is either of the
form 3m or (3m + 1) for some integer m.

Answers

Answered by Anonymous
29

AnSwEr :

  • Let a be the any positive integer which is divided by 3. then we get quotient as q and remainder as r.

Then by Euclid's Division lemma we get :

→ a = 3q + r, 0 ≤ r < 3

⛬ r = 0, 1, 2

Now, putting the value of r we get :

  • When r = 0

⇒a = 3q

Similarly, by taking r = 1 :

⇒a = 3q + 1

Also, for r = 2 :

⇒a = 3q + 2

⛬ Required form number = 3q, 3q + 1 and 3q + 2.

Now,

  • a = 3q

Squaring both the sides we get :

⇒a² = (3q)²

⇒a² = 9q²

⇒a² = 3(3q²)

⇒a² = 3m [∵ 3q² = m]

When,

  • a = 3q + 1

★ Squaring both the sides we get :

⇒a² = (3q + 1)²

⇒a² = (3q)² + 1 + 6q

⇒a² = 9q² + 1 + 6q

⇒a² = 3(3q² + 2q) + 1

⇒a² = 3m + 1 [ 3q² + 2q = m]

  • Hence, square of any positive integer is of the form 3m and 3m + 1.
Answered by VinCus
43

Solution:-

•Using Euclid's division lemma we can prove that the square of any positive integer is either in the form of

Euclid's Division lemma: If two positive integer "a" and "b", and there exists unique integers "q" and "r" such that which satisfies the condition a = bq + r where 0 < r < b

•Let a be the positive integer

Case:

★ a = bq + r ★

•Let b = 3 , r = 0

★ a = 3q + r ★

•Squaring on both sides,

➞ (a)² = (3q)² + 0

➞ (a)² = 9q²

➞ (a)² = 3(3q)²

•Let m stand's for 3q²

➥ a² = 3m ----------》(1)

Case:

★ a = bq + r ★

•Let b = 3 , r = 1

★ a = 3q + 1 ★

•Squaring on both sides,

➞ (a)² = (3q)² +(1)²

➞ (a)² = (3q + 1)²

➞ (a)² = (3q)² + (1)² +2(3q)(1)

➞ (a)² = 9q² + 1 + 6q

➞ (a)² = 3(3q² + 2q) + 1

•Let m stand's for 3q² + 2q

➥ a² = 3m + 1 ----------》(2)

Case:

★ a = bq + r ★

Let b = 3 , r = 2

★ a = 3q + 2 ★

•Squaring on both sides,

➞ (a)² = (3q)² + (2)²

➞ (a)² = (3q + 2)²

➞ (a)² = (3q)² + (2)² + 2(3q)(2)

➞ (a)² = 9q² + 4 + 12q

➞ (a)² = 9q² + 3 + 1 + 12q

➞ (a)² = 3(3q² + 4q + 1) + 1

•Let m stand's for 3q² + 4q + 1

➥ a² = 3m + 1 ----------》(3)

★Therefore , the square of any positive integer is either of the form 3m or 3m+1.


Sitααrα: Marvellous!
VinCus: Thanx!
opmaddy07: Superb!
Anonymous: Your all answers are great :D
VinCus: Nanri Thala!
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