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

$\mathtt{\huge{\underline{\red{Question\:?}}}}$

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

$\mathtt{\huge{\underline{\green{Answer :-}}}}$

$\huge{\bf{\pink{\underline{Given:-}}}}$

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

$\huge{\bf{\orange{\underline{To\:Show:-}}}}$

• The Above Given Statement .

$\huge{\bf{\red{\underline{Proof:-}}}}$

According to the euclids division lemma,

a = bq + r

Where, 6q + r

6q + r 0 ≤ r <6

Consider an given integer a,

Devide the a by 6 , where we have, q our quotient and r our remainder such that,

a = 6q + r,

Where, the value of r is 0,1,2,3,4,5

Case 1 :-

Where r = 0

a = 6q (We get an even no.)

Case 2 :-

Where r = 1

a = 6q + 1 (We get an odd no.)

Case 3 :-

Where, r = 2

a = 6q + 2 (We get an even no.)

Case 4 :-

Where, r = 3

a = 6q + 3 (We get an odd no.)

Case 5 :-

Where, r = 4

a = 6q + 4 (We get an even no.)

Case 6 :-

Where, r = 5,

a= 6q + 5 (We get an odd no.)

Hence , we can say that any positive odd integer is of the form 6q+1 ,6q+3 or 6q+5.

_____________________________________

Answer 2

$\mathfrak{\huge{\underline{\red{To \: Prove :-}}}}$

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

$\mathfrak{\huge{\underline{\green{Proof:-}}}}$

From the euclids division lemma,

$\sf{ ➝\orange{a = bq + r}}$

$\sf{\orange{Where, 6q + r}}$

$\sf{➝\orange{6q + r 0 ≤ r <6}}$

Consider an integer a,

Now,

Divide 'a' by 6 , where : q our quotient and r our remainder .

a = 6q + r,

Where, the value of r is 0,1,2,3,4,5

Case 1 :-

Where r = 0

a = 6q

(We get an even no.)

Case 2 :-

Where r = 1

a = 6q + 1

(We get an odd no.)

Case 3 :-

Where, r = 2

a = 6q + 2

(We get an even no.)

Case 4 :-

Where, r = 3

a = 6q + 3 (We get an odd no.)

Case 5 :-

Where, r = 4

a = 6q + 4 (We get an even no.)

Case 6 :-

Where, r = 5,

a= 6q + 5 (We get an odd no.)

Therefore,$\pink{\underline{ Any\: positive \: odd \: integer \:is \:of \:the \:form \:6q+1 ,6q+3 or 6q+5.}}$

Hence proved

__________________________________