Question

SHOW THAT ANY POSITIVE ODD INTEGER IS OF THE FORM 6q+1 or 6q+3 or 6q+5 WHERE q IS SOME INTEGER

Answer 1

Let a = any positive integer

b = 6

r = 0, 1, 2, 3, 4, 5

According to Euclid's Division Lemma: a = bq + r

So,

a = 6q (It is an even number)

a = 6q + 1 (It is an odd number)

a = 6q + 2 (It is an even number)

a = 6q + 3 (It is an odd number)

a = 6q + 4 (It is an even number)

a = 6q + 5 (It is an odd number)

Therefore, any odd number is in the form 6q + 1, 6q + 3, 6q + 5

Answer 2

$\huge \underline \mathbb {SOLUTION:-}$

Let a be any positive integer and b = 6. Then, by Euclid’s algorithm, a = 6q + r, for some integer q ≥ 0, and r = 0, 1, 2, 3, 4, 5, because 0≤r<6.

Now substituting the value of r, we get,

If r = 0, then a = 6q

Similarly, for r= 1, 2, 3, 4 and 5, the value of a is 6q+1, 6q+2, 6q+3, 6q+4 and 6q+5, respectively.

If a = 6q, 6q+2, 6q+4, then a is an even number and divisible by 2. A positive integer can be either even or odd Therefore, any positive odd integer is of the form of 6q+1, 6q+3 and 6q+5, where q is some integer.