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

Hi friend,


Let a be any positive integer and b = 6.

Then, by Euclids algorithm,

a = 6q + r for some integer q ≥ 0, and r = 0, 1, 2, 3, 4, 5 because 0 ≤ r

Therefore, a = 6q or 6q + 1 or 6q + 2 or 6q + 3 or 6q + 4 or 6q + 5

Also, 6q + 1 = 2 × 3q + 1 = 2k 1 + 1, where k 1 is a positive integer

6q + 3 = (6q + 2) + 1 = 2 (3q + 1) + 1 = 2k 2 + 1, where k 2 is an integer

6q + 5 = (6q + 4) + 1 = 2 (3q + 2) + 1 = 2k 3 + 1, where k 3 is an integer

Clearly, 6q + 1, 6q + 3, 6q + 5 are of the form 2k + 1, where k is an integer.

Therefore, 6q + 1, 6q + 3, 6q + 5 are not exactly divisible by 2. Hence, these expressions of numbers are odd numbers.

And therefore, any odd integer can be expressed in the form 6q + 1, or 6q + 3, or 6q + 5....


Hope this helps you...
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Answer 2

Hi Ankit, Here is the required answer:-

Let the positive odd integer be 'x'.

By Euclid's division lemma,

x=6q+r. (b=6)

So, 0 is equal to less than r .and r is less than 6.

So, R can have the values 0,1,2,3,4,&5.

So Putting values of r,

x = 6q+0. x=6q+4

x=6q+1, x=6q+3

x=6q+2 x=6q+5

But since, we have been given positive odd integer. Therefore, x≠6q, x≠6q+2 & x≠6q+4.

So, x(positive odd integer can be expressed in the form of 6q+1,6q+3 & 6q+5.

HOPE THIS HELPS YOU...
PLEASE MARK THIS ANSWER AS BRAINLIEST IF IT HELPED YOU!!! ☺️☺️