5. Use Euclid's division lemma to show that the cube of any positive integer is of the form
9m, 9m + 1 or 9m +8.
Answers
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Use Euclid's division lemma to show that cube of any positive integer is either of the form 9m, 9m + 1 or 9m+ 8 for some integer 'm'. a3=(3q)3=27q3=9(3q3)=9m where m=3q3 and 'm' is an integer. where m=3q3+3q2+q and 'm' is an integer.9m+8, where m=3q3+6q2+4q and m is an integer.
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Step-by-step explanation:
Let a be any positive integer and b = 3
a = 3q+ r, where q ≥0 and 0≤ r< 3
∴a= 3q or 3q+1 or 3q+2
Therefore, every number can be represented as these three forms. There are three cases
Case 1: When a = 3q,
a³ = (3q)³ = 27q³ = 9(3q³) = 9m
Where m is an integer such that m 3q³
Case 2: When a = 3q+1,
a³= (3q +1)³
a³ = 27q³ + 27q²+ 9q + 1
a³ = 9(3q³ + 3q² + q) + 1
a³= 9m +1
Where m is an integer such that m (3q³ + 3q² + q)
Case 3: When a = 3q + 2,
a³ = (3q +2)³
a³= 27q³ + 54q² + 36q + 8
a³ = 9(3q³+ 6q² + 4q) + 8
a³ = 9m + 8
Where m is an integer such that m = (3q³+6q² + 4q)
Therefore, the cube of any positive integer is of the form 9m, 9m + 1, or 9m + 8.
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