Question

solve it ......................... ​

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.

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Answer 2

Question :

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

Solution :

Let a be any positive integer and $\boxed{\sf{b=2}}$

$\large\sf{\underline{Answer}}$

Then, by By Euclid's algorithm , a=6q+r for some integer q > 0 and r= 0,1,2,3,4,5, or 0 < r < 6 .

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

➡ 6q+0 : 6 is divisible by 2 , so it is an even number.

➡ 6q+1 : 6 is divisible by 2 ,but 1 is not divisible by 2 so it is an odd number.

➡ 6q+2 : 6 is divisible by 2 , and 2 is divisible by 2 so it is an even number.

➡ 6q+3 : 6 is divisible by 2 , but 3 is not divisible by 2 , so it is an odd number.

➡ 6q+4 : 6 is divisible by 2 , and 4 is divisible by 2 so it is an even number.

➡ 6q+5 : 6 is divisible by 2, but 5 is not divisible by 2 so it is an odd number.

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

Hence proved