## Question :_________➡️Use Euclid’s division lemma to show that the square of any positive integer is either of th

Question

Question :
_________

➡️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.
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2 months 2021-07-30T04:48:55+00:00 2 Answers 0 views 0 ### To prove

• The square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

### Proof

Let us consider a positive integer ‘a’

• Divide the positive integer a by 3, and let r be the reminder and b be the quotient such that
• a = 3b + r……………………………(1)
• where r = 0,1,2,3…..

Case 1: Consider r = 0

• Equation (1) becomes
• a = 3b

On squaring both the side

• a^2 = (3b)^2
• a^2 = 9b^2
• a^2 = 3 × 3b^2
• a^2 = 3m
• Where m = 3b^2

Case 2: Let r = 1

• Equation (1) becomes
• a = 3b + 1

Squaring on both the side we get

• a^2 = (3b + 1)^2
• a^2 = (3b)^2 + 1 + (2 × (3b) × 1)
• a^2 = 9b^2 + 6b + 1
• a^2 = 3(3b^2 + 2b) + 1
• a^2 = 3m + 1
• Where m = 3b^2 + 2b

Case 3: Let r = 2

• Equation (1) becomes
• a = 3b + 2

Squaring on both the sides we get

• a^2 = (3b + 2)^2
• a^2 = 9b^2 + 4 + (2 × 3b × 2)
• a^2 = 9b^2 + 12b + 4
• a^2 = 9b^2 + 12b + 3 + 1
• a^2 = 3(3b^2 + 4b + 1) + 1
• a^2 = 3m + 1
• where m = 3b^2 + 4b + 1

∴ square of any positive integer is of the form 3m or 3m+1.