How can i easily solve the question of class 10th maths question the chapter name is real number exercise 1.1. It’s important question of real number. I want to get best solution for this question Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.
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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.
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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.