Question

Show that any positive odd integer is in the form 4q+1 or 4q+3 where q is some integer.

Answer 1

Answer:

I can explain as follows:

When a positive integer is divided by 4, the remainder will be 0, 1, 2, or 3.

By division algorithm, for any positive odd integer there exists integers q and r such that

p = 4q + r where r=0,1,2 or 3.

When r= 0 or 2,

p is divisible by 2 and so p cannot be an odd number i.e. r cannot be equal to 0 or 2.

So, p=4q+1 or p=4q+3.

That is, any positive odd integer is of the form 4q+1 or 4q+3, where q is some integers.

Step-by-step explanation:

Answer 2

$\huge \bf \green{Hey \: there !! }$


$\bf Step-by-step \: explanation:$



Let a be the positive integer.


And, b = 4 .


Then by Euclid's division lemma,


We can write a = 4q + r ,for some integer q and 0 ≤ r < 4 .


°•° Then, possible values of r is 0, 1, 2, 3 .


Taking r = 0 .


a = 4q .



Taking r = 1 .


a = 4q + 1 .


Taking r = 2


a = 4q + 2 .


Taking r = 3 .


a = 4q + 3 .


But a is an odd positive integer, so a can't be 4q , or 4q + 2 [ As these are even ] .



•°• a can be of the form 4q + 1 or 4q + 3 for some integer q .



Hence , it is solved



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