Question

show that any positive odd integer is in the form of 4q±1 or 4q±3 for some integer q​

Answer 1

Answer:

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Answer 2

Answer:

let that we have any positive integer of form 4q+1 or 4q+3

As per Euclid's division lemma

if a and b are two positive integer then

a=bq+r

where r if greater than or equal to 0 but smaller than b

let a positive integer be a and b=4

hence

a=4q

or

a=4q+1

or

a=4q+2

or

a=4q+3

here we observed that 4q and 4q+2 are even integer

the remaining two {4q+1 and 4q+3} are odd positive integer

because integers are of two types

1. odd
2. even

so we can say that every odd positive integer is of form 4q+1 or of 4q+3