Show that the cube of a positive integer of the form 6q + r, q is an integer and r = 0, 1, 2, 3, 4, 5 is also of the form 6m + r
Solution:
Consider a positive integer = 6q + r
where q is an integer and
r = 0, 1, 2, 3, 4, 5.
We know that positive integers are of the form 6q, 6q + 1, 6q + 2, 6q + 3, 6q + 4 and 6q + 5.
By taking cube on both sides
For 6q,
(6q)3 = 216q3
= 6(36q)3 + 0
= 6m + 0, (where m is an integer = (36q)3).
For 6q + 1,
(6q + 1)3 = 216q3 + 108q2 + 18q + 1
= 6(36q3 + 18q2 + 3q) + 1
= 6m + 1, (where m is an integer = 36q3 + 18q2 + 3q).
For 6q + 2,
(6q + 2)3 = 216q3 + 216q2 + 72q + 8
= 6(36q3 + 36q2 + 12q + 1) +2
= 6m + 2, (where m is an integer = 36q3 + 36q2 + 12q + 1).
For 6q + 3,
(6q + 3)3 = 216q3 + 324q2 + 162q + 27
= 6(36q3 + 54q2 + 27q + 4) + 3
= 6m + 3, (where m is an integer = 36q3 + 54q2 + 27q + 4).
For 6q + 4,
(6q + 4)3 = 216q3 + 432q2 + 288q + 64
= 6(36q3 + 72q2 + 48q + 10) + 4
= 6m + 4, (where m is an integer = 36q3 + 72q2 + 48q + 10).
For 6q + 5,
(6q + 5)3 = 216q3 + 540q2 + 450q + 125
= 6(36q3 + 90q2 + 75q + 20) + 5
= 6m + 5, (where m is an integer = 36q3 + 90q2 + 75q + 20).
Therefore, the cube of a positive integer of the form 6q + r, q is an integer and r = 0, 1, 2, 3, 4, 5 is also of the form 6m + r.
✦ Try This: Show that the cube of a positive integer of the form 7q + r, q is an integer and r = 0, 1, 2, 3, 4, 5 is also of the form 7m + r
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 1
NCERT Exemplar Class 10 Maths Exercise 1.4 Problem 1
Show that the cube of a positive integer of the form 6q + r, q is an integer and r = 0, 1, 2, 3, 4, 5 is also of the form 6m + r
Summary:
The cube of a positive integer of the form 6q + r, q is an integer and r = 0, 1, 2, 3, 4, 5 is also of the form 6m + r. Hence proved
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