Math, asked by akashsaha6020, 9 months ago

Show that any positive odd integer is of the from 4q+1 or 4q+3 where q is some integer

Answers

Answered by Anonymous

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let a be any positive integer

by EDL a = bq +r

0 ≤ r < b

possible remainders are 0, 1, 2 , 3

this shows that a can be in the form of 4q, 4q+1, 4q+2, 4q+3 q is quotient

as a is odd a can't be the form of 4q or 4q+2 as they are even

so a ill be in the form of 4q + 1 or 4q+3

hence proved

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Answered by Sudhir1188

Let 'n' be a positive integer which is Divided by 4 . We get some quotient 'q' and some remainder 'r'.

By Euclid Division :

=> n = 4q+r

Where r = 0 ,1,2,3

Putting r = 0

=> n = 4q

=> n = 2(2q) . [Divisible by 2]

Putting r = 1

=> n = 4q+1 (It is odd)

Putting r = 2

=> n = 4q+2

=> n = 2(2q+1). [Divisible by 2]

Putting r = 3

=> n = 4q+3. [ Not Divisible by 2]

So every positive odd integer is of the from 4q+1 or 4q+3 where q is some integer.

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