Question

Show that every positive odd integer is of the form 6q+1, 6q+3 or 6q+5, where q is some positive integer.

Answer 1

Hope you get the solution......

Answer 2

Answer:

Data - Positive odd integer

To prove - Form is 6q + 1, 6q +3 or 6q + 5.

Proof - Let a, b = 6 such that a > b.

Applying Euclid's Division Lemma,

➸ a = bq + r, 0 ≤ r < b

➸ a = 6q + r, 0 ≤ r < 6

➸ Hence, r = 0, 1, 2, 3, 4, 5, 6.

When r = 0,

➸ a = 6q + 0 = 6q .....(1) [Even]

➸ a = 6q + 1 .....(2) [Odd]

➸ a = 6q + 2 .....(3) [Even]

➸ a = 6q + 3 .....(4) [Odd]

➸ a = 6q + 4 .....(5) [Even]

➸ a = 6q + 5 .....(6) [Odd]

Hence, From (2), (4) and (6) we can conclude that any positive odd integer is in the form of 6q + 1, 6q +3 or 6q + 5.

What does euclid's lemma state ?

For every given integers a and b, there exists two unique integers q and r that satisfies a = bq + r, 0 ≤ r < b.