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Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.
Dear StudentLet x be any positive integer and y = 3.By Euclid’s division algorithm, then,x = 3q+r,where q≥0 and r = 0, 1, 2, as r ≥ 0 and r < 3.Therefore, putting the value of r,we get,x = 3qorx = 3q + 1orx = 3q + 2Now, by taking the cube of all the three above expressions, we get,Case (i):When r = 0, then,x^2= (3q)^3= 27q^3= 9(3q^3)= 9m;where m = 3q^3Case (ii):When r = 1, then,x^3= (3q+1)^3= (3q)^3+1^3+3×3q×1(3q+1) = 27q^3+1+27q^2+9qTaking 9 as common factor, we get,x^3= 9(3q^3+3q^2+q)+1Putting(3𝑞^3+ 3𝑞^2+ 𝑞)= m, we get,Putting (3q^3+3q^2+q) = m, we get ,x^3= 9m+1Case (iii): When r = 2, then,x^3= (3q+2)^3= (3q)^3+2^3+3×3q×2(3q+2) = 27q^3+54q^2+36q+8Taking 9 as common factor, we get,x^3=9(3q^3+6q^2+4q)+8Putting (3q^3+6q^2+4q) = m,we get ,x^3= 9m+8Therefore, from all the three cases explained above,it is proved that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.Thanks
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