Question

Show that any odd positive integers is in the former of 4k+1 or 4k+3,k €z

Answer 1

To show that

any odd positive integers is in the former of 4k+1 or 4k+3, k ∈z

Let a be any positive integer

From Euclid's Division lemma, a can be written as

$a=bk+r$

Where $0\le r<b$

If we take $b=4$

Then the possible values of r will be 0, 1, 2, 3

For $r=0$

$a=4k$ which is an even integer

For $r=1$

$a=4k+1$ which is an odd integer

For $r=2$

$a=4k+2$ which is an even integer

For $r=3$

$a=4k+3$ which is an odd integer

Therefore, odd positive integers are of the form $4k+1$ and $4k+3$

Hope this helps.

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Answer 2

Answer:  The proof is done below.

Step-by-step explanation:   We are given to show that any positive odd integer is of the form 4k + 1 or 4k + 3.

Let n be any positive integer

Euclid's algorithm :  if a and b are two positive integers, there exist unique integers q and r satisfying

a = bq + r, where 0 ≤ r < b.

Let us consider that b = 4. Then, we have

n = 4q + r, where 0 ≤ r < 4

Since 0 ≤ r < 4, the possible remainders are 0, 1, 2 and 3.

So, n can be of the form 4q, 4q + 1, 4q + 2 and 4q + 3.

Since the number n is odd, so a cannot be equal to 4q or 4q + 2, because they are both divisible by 2.

Therefore, the number a is of the form 4q + 1 or 4q + 3.

Thus, any odd positive integer n the form 4k+1 or 4k+3.

Hence showed.