Class 10 Mathematics - Chapter Real Numbers NCERT Solutions | Use Euclid’s division lemma to sho
Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.
Let a be any positive integer and b = 3. Then, by euclid’s algorithm
a = 9q + r, where q ≥ 0 and r = 0, 1, 2 [ 0 ≤ r ≤ b ]
For r = 0, x = 3q, or
For r = 1, x = 3q +1
For r = 2, x = 3q + 2
Now by taking the cube of all the three above terms, we get,
Case (i) : when r = 0, then,
x3 = (3q)3 = 27q3 = 9 (3q3) = 9m; where m = 3q
Case (ii) : when r = 1, then,
x3 = (3q +1)3 = (3q)3 + 13 +3 × 3q × 1 (3q + 2 ) = 27q3 + 1 +27q2 + 9q
Taking 9 as common factor, we get,
x3 = 9 (3q3 +3q2 + q) +1
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
x = 9 (3q + 6q + 4q) + 8
Putting (3q + 6q + 4q) = m, we get,
x = 9m + 8,
Therefore, it is proved that the cube of any positive integer is of the form 9m, 9m + 1, 9m + 8.
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