use euclied's dividion lemma to show that the cube of any positive integer is of the from 9m,9m+1 or 9m+8.
Answers
Step-by-step explanation:
To prove that,
The cube of any positive integer is of the form 9m, 9m+1 or 9m+8.
First of all,
Let's assume that a be any positive integer and b = 3.
Therefore, we can write,
=> a = 3q + r,
Where q ≥ 0 and 0 ≤ r < 3.
Therefore, we have possible values of r = 0,1,2
Now, we know that,
Every number can be represented as these three forms.
Therefore, we have 3 cases.
Case I: When a = 3q,
=> a^3 = (3q)^3
=> a^3 = 27q^3
=> a^3 = 9(3q^3)
=> a^3 = 9m
Where, m is an integer such that m = 3q^3.
Case II: When a = 3q + 1,
=> a^ 3 = (3q +1)^ 3
=>a^3 = 27q^ 3 + 27q ^2 + 9q + 1
=> a^ 3 = 9(3q^ 3 + 3q ^2 + q) + 1
=> a ^3 = 9m + 1
Where, m is an integer such that m = (3q ^3 + 3q^ 2+ q).
Case III: When a = 3q + 2,
=> a^3 = (3q +2)^ 3
=> a^ 3 = 27q^ 3 + 54q ^2 + 36q + 8
=> a^ 3 = 9(3q ^3 + 6q ^2 + 4q) + 8
=> a^ 3 = 9m + 8
Where, m is an integer such that m = (3q^ 3 + 6q^ 2+ 4q).
Hence, the cube of any positive integer is of the form 9m, 9m + 1, or 9m + 8.
Thus, Proved.