Question

which equally division Lemma to show that the cube of any positive integer is of the form 9M 9M + 1 or 9 M + 8 ​

Answer 1

Heyaa☺

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Let,

m be any positive integer that is 3m,3m + 1 and 3m+2.

To prove that the cube of these can be written as 9q,9q+1 and 9q+8.

Now,

(3m)^3 =》 27m^3 =9(3m^3)=9q where,q = 3m^3.

(3m+1)^3 = 27m^3 +27m^2+ 9m +1=9(3m^3+3m^2+m)+1=9q + 1

(3m+2)^3 = 27m^3 + 54m^2+36m+8=9(3m^3 + 6m^2 + 4m)+8.=9q+8,Where q=3m^3 + 6m^2 +4m.

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Hope you understand...✌

Now,

(3m)^3