Question

Show that any positive integer is always in the form of 6q, 6q+2, 6q+4, if it is even and 6q+1, 6q+3, 6q+5 if it is odd.?

Answer 1

Step-by-step explanation:

Assumption:

Let a be any positive integer and b = 6.

By using Euclid's division lemma to 6q we obtain,

a = 6q + 0, where 0 ≤ r < 6

Therefore, remainder can be r = 0, 1, 2, 3, 4, or 5.

Taking r's value,

i) a = 6q + 0 = 6q

ii) a = 6q + 1

iii) a = 6q + 2

iv) a = 6q + 3

v) a = 6q + 4

vi) a = 6q + 5

But, 6q, 6q + 2 and 6q + 4 gives even value of a for some integer q.

Therefore, any positive even integer is of the form of 6q, 6q + 2 and 6q + 4 for some integer q.

But, 6q + 1, 6q + 3 and 6q + 5 gives odd value of a for some integer q.

Therefore, any positive odd integer is of the form of 6q + 1, 6q + 3 and 6q + 5 for some integer q.

Hence, showed.

Answer 2

Answer:

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