Show that cube of a positive integer of the form 6q+r, where q is an integer and r = 0,1,2,3,4,5 is also of the form 6m+r
Answers
Answer:
6q + r is a positive integer, where q is an integer and r = 0, 1, 2, 3, 4, 5
Then, the positive integers are of the form 6q, 6q+1, 6q+2, 6q+3, 6q+4 and 6q+5.
Taking cube of each term, we have,
(6q)3 = 216 q3 = 6(36q)3 + 0
= 6m + 0, where m is an integer
(6q+1)3 = 216q3 + 108q2 + 18q + 1
= 6(36q3 + 18q2 + 3q) + 1
= 6m + 1, where m is an integer
(6q+2)3 = 216q3 + 216q2 + 72q + 8
= 6(36q3 + 36q2 + 12q + 1) +2
= 6m + 2, where m is an integer
(6q+3)3 = 216q3 + 324q2 + 162q + 27
= 6(36q3 + 54q2 + 27q + 4) + 3
= 6m + 3, where m is an integer
(6q+4)3 = 216q3 + 432q2 + 288q + 64
= 6(36q3 + 72q2 + 48q + 10) + 4
= 6m + 4, where m is an integer
(6q+5)3 = 216q3 + 540q2 + 450q + 125
= 6(36q3 + 90q2 + 75q + 20) + 5
= 6m + 5, where m is an integer
Hence, 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.
Answer:
6q + r is a positive integer, where q is an integer and r = 0, 1, 2, 3, 4, 5 Then, the positive integers are of the form 6q, 6q+1, 6q+2, 6q+3, 6q+4 and 6q+5. Taking cube of each term, we have, (6q)3 = 216 q3 = 6 × 36 q 3 6×36q3 + 0 = 6m + 0, where m is an integer (6q+1)3 = 216q3 + 108q2 + 18q + 1 = 6(36q3 + 18q2 + 3q) + 1 = 6m + 1, where m is an integer (6q+2)3 = 216q3 + 216q2 + 72q + 8 = 6(36q3 + 36q2 + 12q + 1) +2 = 6m + 2, where m is an integer (6q+3)3 = 216q3 + 324q2 + 162q + 27 = 6(36q3 + 54q2 + 27q + 4) + 3 = 6m + 3, where m is an integer (6q+4)3 = 216q3 + 432q2 + 288q + 64 = 6(36q3 + 72q2 + 48q + 10) + 4 = 6m + 4, where m is an integer (6q+5)3 = 216q3 + 540q2 + 450q + 125 = 6(36q3 + 90q2 + 75q + 20) + 5 = 6m + 5, where m is an integer Hence, 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.Read more on Sarthaks.com - https://www.sarthaks.com/12821/show-that-the-cube-of-positive-integer-of-the-form-6q-is-an-integer-and-5-is-also-of-the-form-6m