Answer

Verified

381.6k+ 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

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

How do you graph the function fx 4x class 9 maths CBSE

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

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

Difference Between Plant Cell and Animal Cell

Why is there a time difference of about 5 hours between class 10 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Draw a labelled sketch of the human eye class 12 physics CBSE