Use Euclid's division lemma to sh
that the cube of a positive integer in
the form 9m, 9m+1 or 9m+8.
(OR)
Show that the cube of any positive inis
ger is of form 9m or 9m + 1 or 9m +
where m is an integer.
Answers
Step-by-step explanation:
let, 3 numbers be 3m,3m+1,3m+2.
so, (3m)^3=9(3m)^3 (by simplification) =9q
(3m+1)^3=9(3m^3+3m^2+m) =9q+1
(3m+2)^3=9q+8
::q is some variables...
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Answer:
Step-by-step explanation:
Let a be any positive integer and b = 3
a = 3q + r, where q ≥ 0 and 0 ≤ r < 3
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) ]
∴ the cube of any positive integer is of the form 9m, 9m + 1, or 9m + 8.