Use Euclid’s division lemma to show that the cube. of any positive integer is of the form 9m,9m+1 or 9m+8 . About the author Caroline
[tex] \: \huge\bf\underline \red{\underline{Answer}}[/tex] Let a be any positive integer and b = 3 a = 3q + r, where q ≥ 0 and 0 ≤ r < 3 [tex]∴ r = 0,1,2 [/tex] Therefore, every number can be represented as these three forms. There are three cases. Case 1: When a = 3q, Where m is an integer such that m = [tex](3q)³ = 27q³ [/tex] [tex] = > 9(3q³) = 9m [/tex] Case 2: When a = 3q + 1, [tex]a = (3q +1) ³ [/tex] [tex] = > a = 27q ³+ 27q ² + 9q + 1 [/tex] [tex] = > a = 9(3q ³ + 3q ² + q) + 1[/tex] [tex] = > a = 9m + 1[/tex] [tex]( Where \: m = 3q³ + 3q² + q )[/tex] Case 3: When a = 3q + 2, [tex]a = (3q +2) ³ [/tex] [tex] = > a = 27q³ + 54q² + 36q + 8 [/tex] [tex] = > a = 9(3q³ + 6q² + 4q) + 8[/tex] [tex] = > a = 9m + 8[/tex] Where m is an integer such that m = [tex] (3q³ + 6q² + 4q) [/tex] Therefore, the cube of any positive integer is of the form 9m, 9m + 1, or 9m + 8. Reply
❥คɴᎦᴡєя Let us consider a and b where a be any positive number and b is equal to 3. According to Euclid’s Division Lemma a = bq + r So r is an integer greater than or equal to 0 and less than 3. Hence r can be either 0, 1 or 2. Case 1: When r = 0, the equation becomes a = 3q Cubing both the sides a3 = (3q)3 a3 = 27 q3 a3 = 9 (3q3) a3 = 9m where m = 3q3 Case 2: When r = 1, the equation becomes a = 3q + 1 Cubing both the sides a3 = (3q + 1)3 a3 = (3q)3 + 13 + 3 × 3q × 1(3q + 1) a3 = 27q3 + 1 + 9q × (3q + 1) a3 = 27q3 + 1 + 27q2 + 9q a3 = 27q3 + 27q2 + 9q + 1 a3 = 9 ( 3q3 + 3q2 + q) + 1 a3 = 9m + 1 Where m = ( 3q3 + 3q2 + q) Case 3: When r = 2, the equation becomes a = 3q + 2 Cubing both the sides a3 = (3q + 2)3 a3 = (3q)3 + 23 + 3 × 3q × 2 (3q + 1) a3 = 27q3 + 8 + 54q2 + 36q a3 = 27q3 + 54q2 + 36q + 8 a3 = 9 (3q3 + 6q2 + 4q) + 8 a3 = 9m + 8 Where m = (3q3 + 6q2 + 4q)therefore a can be any of the form 9m or 9m + 1 or, 9m + 8. Reply
[tex] \: \huge\bf\underline \red{\underline{Answer}}[/tex]
Let a be any positive integer and b = 3
a = 3q + r, where q ≥ 0 and 0 ≤ r < 3
[tex]∴ r = 0,1,2 [/tex]
Therefore, every number can be represented as these three forms. There are three cases.
Case 1:
When a = 3q,
Where m is an integer such that m =
[tex](3q)³ = 27q³ [/tex]
[tex] = > 9(3q³) = 9m [/tex]
Case 2:
When a = 3q + 1,
[tex]a = (3q +1) ³ [/tex]
[tex] = > a = 27q ³+ 27q ² + 9q + 1 [/tex]
[tex] = > a = 9(3q ³ + 3q ² + q) + 1[/tex]
[tex] = > a = 9m + 1[/tex]
[tex]( Where \: m = 3q³ + 3q² + q )[/tex]
Case 3:
When a = 3q + 2,
[tex]a = (3q +2) ³ [/tex]
[tex] = > a = 27q³ + 54q² + 36q + 8 [/tex]
[tex] = > a = 9(3q³ + 6q² + 4q) + 8[/tex]
[tex] = > a = 9m + 8[/tex]
Where m is an integer such that m =
[tex] (3q³ + 6q² + 4q) [/tex]
Therefore, the cube of any positive integer is of the form 9m, 9m + 1, or 9m + 8.
❥คɴᎦᴡєя
Let us consider a and b where a be any positive number and b is equal to 3.
According to Euclid’s Division Lemma
a = bq + r
So r is an integer greater than or equal to 0 and less than 3.
Hence r can be either 0, 1 or 2.
Case 1: When r = 0, the equation becomes
a = 3q
Cubing both the sides
a3 = (3q)3
a3 = 27 q3
a3 = 9 (3q3)
a3 = 9m
where m = 3q3
Case 2: When r = 1, the equation becomes
a = 3q + 1
Cubing both the sides
a3 = (3q + 1)3
a3 = (3q)3 + 13 + 3 × 3q × 1(3q + 1)
a3 = 27q3 + 1 + 9q × (3q + 1)
a3 = 27q3 + 1 + 27q2 + 9q
a3 = 27q3 + 27q2 + 9q + 1
a3 = 9 ( 3q3 + 3q2 + q) + 1
a3 = 9m + 1
Where m = ( 3q3 + 3q2 + q)
Case 3: When r = 2, the equation becomes
a = 3q + 2
Cubing both the sides
a3 = (3q + 2)3
a3 = (3q)3 + 23 + 3 × 3q × 2 (3q + 1)
a3 = 27q3 + 8 + 54q2 + 36q
a3 = 27q3 + 54q2 + 36q + 8
a3 = 9 (3q3 + 6q2 + 4q) + 8
a3 = 9m + 8
Where m = (3q3 + 6q2 + 4q)therefore a can be any of the form 9m or 9m + 1 or, 9m + 8.