Show that every positive odd integer is of the form 4q+1 or 4q+3

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Finally after watching 100 videos i found the best one from which i could understand

parvhadhantaparvhadhanta
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Sir can I solve like this

Let the positive integers be a, and b = 8 . Now by Euclid's division lemma:-
a = bq + r. Where 0 < r < b

Case 1 :- when r = 0
a = 8q + 0
= 4 ( 2q ). Even
4q

Case 2 :- when r = 1
a = 8q + 1
= 4 ( 2q ) + 1. Odd
4q + 1

Case 3:- when r = 2
a = 8q + 2
= 4( 2q ) + 2. Even
4q + 2

Case 4 :- when r = 3
a= 8q + 3
= 4( 2q ) + 3. Odd
4q + 3

Case 5:- when r = 4
a= 8q + 4
= 4 ( 2q + 1 ). Odd
4q

Case 6 :- when r = 5
a = 8q + 5
= 8q + 4+ 1
= 4 ( 2q +1 ) +1 odd
4q +1

Case 7 :- when r = 6
a= 8q + 6
= 4 ( 2q +1 ) + 2. Even
4q + 2

Case 8 :- when r = 7
a = 8q + 7
= 4 ( 2q + 1 ) + 3. Odd
= 4q + 3

Is it correct ☝️

khushbushrivas
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Best explanation sir but I haven't understood last how to find odd & even. Integers 😓

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