This is the basic and conceptual question from real numbers in which we have been asked to prove that any positive odd integer is of the form (4m+1) or (4m+3), where m is some integer.
RS Aggarwal, Class 10, chapter 1A, question no 7
Rajan@2021Guru
Show that any positive odd integer is of the form (4m+1) or (4m+3), where m is some integer.
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Let s be any positive integer.
On dividing s by 4, let m be the quotient and r be the remainder
By Euclid’s division lemma,
s=4m+r, where 0≤r<4
So we have ,s=4m or s=4m+1 or s=4m+2 or s=4m+3.
here 4m,4m+2 are multiples of 2, which revert to even values of s.
again , s=4m+1 or s=4m+3 are odd values of s.
Thus, any positive odd integer is of the form (4m+1) or (4m+3) where s is any odd integer.