Question

solve this for jackpot​

Answer 1

SOLUTIONS

=) 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.

hope it helps

Answer 2

Refer to the attachment!

Thanks!

all the best for your exams.