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

Last updated date: 19th Jul 2024
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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.