Question

Use division algorithm to 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

Step-by-step explanation:

Given :-

Any Positive Odd integer

To find:-

Use division algorithm to show that any positive odd integer is of the form 6q + 1,or 6q + 3 or 6q + 5, where q is some integer.

Solution:-

We know that

Euclid's Division Lemma:-

For any two positive integers there exists two positive integers q and r satisfying a=bq+r,0≤r<b.

Let consider b= 6 then

a = 6q+r , 0<r<b

The possible values of r = 0,1,2,3,4,5.

I) If r = 0 then a = 6q+0

=> a = 6q--------------(1)

=> a is an even number.

ii) If r = 1 then

=>a = 6q +1-----------(2)

=> a is an odd number

iii) If r = 2 then

=>a = 6q+2 -----------(3)

=> a is an even number

iv) If r = 3 then

=> a = 6q+3---------(4)

=>a is an odd number.

v) If r = 4 then

a = 6q+4-----------(5)

=> a is an even number

vi) If r = 5 then

=> a =6q+5 -------(6)

=> a is an odd number.

From (2),(4)&(6)

a is an odd integer.

Answer:-

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

Used formula:-

Euclid's Division Lemma:-

For any two positive integers there exists two positive integers q and r satisfying a=bq+r,0≤r<b.