use Euclid’s division lemma to show that the square of any positive integer is either of the from 3m or 3m + 1 for some integer m.
Answers
Answer:
Let 'a' be any positive integer.
On dividing it by 3 , let 'q' be the quotient and 'r' be the remainder.
Such that ,
a = 3q + r , where r = 0 ,1 , 2
When, r = 0
∴ a = 3q
When, r = 1
∴ a = 3q + 1
When, r = 2
∴ a = 3q + 2
When , a = 3q
On squaring both the sides,
\begin{gathered}{a}^{2} = 9 {q}^{2} \\ {a}^{2} = 3 \times (3 {q}^{2} ) \\ {a}^{2} = 3 \\ where \: m = 3 {q}^{2}\end{gathered}
a
2
=9q
2
a
2
=3×(3q
2
)
a
2
=3
wherem=3q
2
When, a = 3q + 1
On squaring both the sides ,
\begin{gathered}{a}^{2} = (3q + 1)^{2} \\ {a}^{2} = 9 {q}^{2} + 2 \times 3q \times 1 + {1}^{2} \\ {a}^{2} = 9 {q}^{2} + 6q + 1 \\ {a}^{2} = 3(3 {q}^{2} + 2q) + 1 \\ {a}^{2} = 3m + 1 \\ where \: m \: = 3 {q}^{2} + 2q\end{gathered}
a
2
=(3q+1)
2
a
2
=9q
2
+2×3q×1+1
2
a
2
=9q
2
+6q+1
a
2
=3(3q
2
+2q)+1
a
2
=3m+1
wherem=3q
2
+2q
When, a = 3q + 2
On squaring both the sides,
\begin{gathered}{a}^{2} = (3q + 2)^{2} \\ {a}^{2} = 3 {q}^{2} + 2 \times 3q \times 2 + {2}^{2} \\ {a}^{2} = 9 {q}^{2} + 12q + 4 \\ {a}^{2} = (9 {q}^{2} + 12q + 3) + 1 \\ {a}^{2} = 3(3 {q}^{2} + 4q + 1) + 1 \\ {a}^{2} = 3m + 1 \\ where \: m \: = 3 {q}^{2} + 4q + 1\end{gathered}
a
2
=(3q+2)
2
a
2
=3q
2
+2×3q×2+2
2
a
2
=9q
2
+12q+4
a
2
=(9q
2
+12q+3)+1
a
2
=3(3q
2
+4q+1)+1
a
2
=3m+1
wherem=3q
2
+4q+1
Therefore , the square of any positive integer is either of the form 3m or 3m+1.
Answer:
Let us consider a positive integer a
Let us consider a positive integer aDivide the positive integer a by 3, and let r be the reminder and b be the quotient such that
a = 3b + r……………………………(1)
a = 3b + r……………………………(1)where r = 0,1,2,3…..
a = 3b + r……………………………(1)where r = 0,1,2,3…..Case 1: Consider r = 0
a = 3b + r……………………………(1)where r = 0,1,2,3…..Case 1: Consider r = 0Equation (1) becomes
a = 3b + r……………………………(1)where r = 0,1,2,3…..Case 1: Consider r = 0Equation (1) becomesa = 3b
On squaring both the side
On squaring both the sidea2 = (3b)2
On squaring both the sidea2 = (3b)2a2 = 9b2
On squaring both the sidea2 = (3b)2a2 = 9b2a2 = 3 × 3b2
On squaring both the sidea2 = (3b)2a2 = 9b2a2 = 3 × 3b2a2 = 3m
On squaring both the sidea2 = (3b)2a2 = 9b2a2 = 3 × 3b2a2 = 3mWhere m = 3b2
On squaring both the sidea2 = (3b)2a2 = 9b2a2 = 3 × 3b2a2 = 3mWhere m = 3b2Case 2: Let r = 1
On squaring both the sidea2 = (3b)2a2 = 9b2a2 = 3 × 3b2a2 = 3mWhere m = 3b2Case 2: Let r = 1Equation (1) becomes
a = 3b + 1
a = 3b + 1Squaring on both the side we get
a = 3b + 1Squaring on both the side we geta2 = (3b + 1)2
a = 3b + 1Squaring on both the side we geta2 = (3b + 1)2a2 = (3b)2 + 1 + 2 × (3b) × 1
a = 3b + 1Squaring on both the side we geta2 = (3b + 1)2a2 = (3b)2 + 1 + 2 × (3b) × 1a2 = 9b2 + 6b + 1
a = 3b + 1Squaring on both the side we geta2 = (3b + 1)2a2 = (3b)2 + 1 + 2 × (3b) × 1a2 = 9b2 + 6b + 1a2 = 3(3b2 + 2b) + 1
a2 = 3m + 1
a2 = 3m + 1Where m = 3b2 + 2b
a2 = 3m + 1Where m = 3b2 + 2bCase 3: Let r = 2
a2 = 3m + 1Where m = 3b2 + 2bCase 3: Let r = 2Equation (1) becomes
a2 = 3m + 1Where m = 3b2 + 2bCase 3: Let r = 2Equation (1) becomesa = 3b + 2
a2 = 3m + 1Where m = 3b2 + 2bCase 3: Let r = 2Equation (1) becomesa = 3b + 2Squaring on both the sides we get
a2 = 3m + 1Where m = 3b2 + 2bCase 3: Let r = 2Equation (1) becomesa = 3b + 2Squaring on both the sides we geta2 = (3b + 2)2
a2 = 3m + 1Where m = 3b2 + 2bCase 3: Let r = 2Equation (1) becomesa = 3b + 2Squaring on both the sides we geta2 = (3b + 2)2a2 = 9b2 + 4 + (2 × 3b × 2)
a2 = 3m + 1Where m = 3b2 + 2bCase 3: Let r = 2Equation (1) becomesa = 3b + 2Squaring on both the sides we geta2 = (3b + 2)2a2 = 9b2 + 4 + (2 × 3b × 2)a2 = 9b2 + 12b + 3 + 1
a2 = 3m + 1Where m = 3b2 + 2bCase 3: Let r = 2Equation (1) becomesa = 3b + 2Squaring on both the sides we geta2 = (3b + 2)2a2 = 9b2 + 4 + (2 × 3b × 2)a2 = 9b2 + 12b + 3 + 1a2 = 3(3b2 + 4b + 1) + 1
a2 = 3m + 1Where m = 3b2 + 2bCase 3: Let r = 2Equation (1) becomesa = 3b + 2Squaring on both the sides we geta2 = (3b + 2)2a2 = 9b2 + 4 + (2 × 3b × 2)a2 = 9b2 + 12b + 3 + 1a2 = 3(3b2 + 4b + 1) + 1a2 = 3m + 1
a2 = 3m + 1Where m = 3b2 + 2bCase 3: Let r = 2Equation (1) becomesa = 3b + 2Squaring on both the sides we geta2 = (3b + 2)2a2 = 9b2 + 4 + (2 × 3b × 2)a2 = 9b2 + 12b + 3 + 1a2 = 3(3b2 + 4b + 1) + 1a2 = 3m + 1where m = 3b2 + 4b + 1
a2 = 3m + 1Where m = 3b2 + 2bCase 3: Let r = 2Equation (1) becomesa = 3b + 2Squaring on both the sides we geta2 = (3b + 2)2a2 = 9b2 + 4 + (2 × 3b × 2)a2 = 9b2 + 12b + 3 + 1a2 = 3(3b2 + 4b + 1) + 1a2 = 3m + 1where m = 3b2 + 4b + 1∴ square of any positive integer is of the form 3m or 3m+1.
square of any positive integer is of the form 3m or 3m+1.Hence proved.