Use Euclid’s division lemma to show that the cube. of any positive integer is of the form 9m,9m+1 or 9m+8​

2 thoughts on “Use Euclid’s division lemma to show that the cube. of any positive integer is of the form 9m,9m+1 or 9m+8​”

1. ★Important formula★

   • (a+b)³= a³+3a²b+3ab²+b³

   Reply

2. 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.

   Reply