Question

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

Answer 1

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let a be any positive integer

then

b= 4

a= bq+r

0≤r<b

0≤r<4

r= 0,1,2,3

case 1.

r=0

a= bq+r

4q+0

4q

case 2.

r=1

a= 4q+1

6q+1

case3.

r=2

a=4q+2

case 4.

r=3

a=4q+3

hence from above it is proved that any positive integer is of the form 4q, 4q+1,4q+2,4q+3

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Answer 2

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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