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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.

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.

Grade:12th pass

1 Answers

Pawan Prajapati
askIITians Faculty 8741 Points
2 months ago
Solution: Let 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 = 3q or x = 3q + 1 or x = 3q + 2 Now, by taking the cube of all the three above expressions, we get, Case (i): When r = 0, then, x2= (3q)3 = 27q3= 9(3q3)= 9m; where m = 3q3 Case (ii): When r = 1, then, x3 = (3q+1)3 = (3q)3 +13+3×3q×1(3q+1) = 27q3+1+27q2+9q Taking 9 as common factor, we get, x3 = 9(3q3+3q2+q)+1 Putting = m, we get, Putting (3q3+3q2+q) = m, we get , x3 = 9m+1 Case (iii): When r = 2, then, x3 = (3q+2)3= (3q)3+23+3×3q×2(3q+2) = 27q3+54q2+36q+8 Taking 9 as common factor, we get, x3=9(3q3+6q2+4q)+8 Putting (3q3+6q2+4q) = m, we get , x3 = 9m+8 Therefore, 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.

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