Answer

Verified

348.3k+ views

**Hint:**In this question we are going to prove any odd integer is of the form \[4q + 1\] or \[4q + 3\]. To prove this we are going to use “Euclid’s Division Lemma”. Euclid’s Division Lemma states that, given positive integers \[a\] and \[b\], there exist unique integers \[q\] and \[r\] satisfying \[a = bq + r{\text{,}}0 \leqslant r < b\].

**Complete step-by-step solution:**

Here, we take \[b = 4\] because as per our question we want to prove is of the form \[4q + 1\] or \[4q + 3\],

Let \[a\] be any positive integer and \[b = 4\].

Here, the integer is \[4\]so we consider \[b = 4\].

As per Euclid’s Division Lemma,

\[a = 4q + r\], for some integer \[q \geqslant 0\] and \[r = 0{\text{,}}1,2,3\] because \[0 \leqslant r < 4\].

Now substituting the value of \[r\], we get,

If \[r = 0\], then \[a = 4q\]

Similarly, for \[r = 1,2\] and \[3\], the value of \[a\] is, \[a = 4q + 1\], \[a = 4q + 2\] and \[a = 4q + 3\] respectively.

If \[a = 4q\] and \[a = 4q + 2\] then \[a\] is an even number and divisible by \[2\]. A positive integer can be either even or odd.

Therefore, any positive odd integer is of the form \[4q + 1\] or \[4q + 3\], where q is some integer.

**Note:**Euclid’s division algorithm is a technique to compute the Highest Common Factor (HCF) of two given positive integers. HCF of two positive integers \[a\] and \[b\] is the largest positive integer \[d\] that divides both \[a\] and \[b\]. Euclid’s division algorithm is based on Euclid’s Division Lemma.

Euclid’s Division Lemma has many applications related to divisibility of integers. It can be used to find the HCF of two numbers. The process of finding the HCF of two numbers using Euclid’s Division Lemma is called Euclid’s Division Algorithm.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Difference Between Plant Cell and Animal Cell

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Change the following sentences into negative and interrogative class 10 english CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Select the word that is correctly spelled a Twelveth class 10 english CBSE