Question

show that any positive odd integer is of the form 4q+1 or 4q+3 where q is some integer​

Answer 1

Answer:

Step-by-step explanation:

we know that 4 is an even integer and multiples of 4 are also even intigers so when we add odd number to an even number we get odd number so let q be any integer and 4q is an even integer and so 4q+1 is odd integer as 4q is even and we are adding 1 to even integer similiarly for 4q+3

4q=even

4q+1=odd(as shown above) as even number +odd number

4q=even

4q+3=(odd) as even number +odd number

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 .