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  • Show that every positive odd integer is of the form (4q+1) or (4q+3), where q is some integer.    

Show that every positive odd integer is of the form (4q+1) or (4q+3), where q is some integer.

 

 

Answers (1)

From the euclid's theorem,

a=bq+r where b is divisor q is quotient and r is the remainder and 0\leq r\leq b.

Let's assume value of b =4

then a=4q+r,0\leq r\leq 4

Since divisor is 4, the possible remainder for the condition are 0,1,2,3.

So the possible values of a are,

4q,4q+1,4q+2,4q+3

Since a is odd, a cannot be 4q and 4q+2 because both are divisible by 2.

So the final values of a are 4q+1 and 4q+3.

 Hence all positive odd numbers are of the form 4q+1 and 4q+3.

 

Hence Proved.

Posted by

Safeer PP

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