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._______________________ ❌⚠️Don’t spam pleech ⚠️❌ About the author Alice
Answer: [tex]{\huge{\blue{\texttt{\orange A\red N\green S\pink W\blue E\purple R\red}}}}[/tex] 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. Hence proved. Done ✅ Reply
Answer:
[tex]{\huge{\blue{\texttt{\orange A\red N\green S\pink W\blue E\purple R\red}}}}[/tex]
To prove
Proof
Let us consider a positive integer ‘a’
Case 1: Consider r = 0
On squaring both the side
Case 2: Let r = 1
Squaring on both the side we get
Case 3: Let r = 2
Squaring on both the sides we get
∴ square of any positive integer is of the form 3m or 3m+1.
Hence proved.
Done ✅
Answer:
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