Question

show that every positive odd integer is of the form 4q+1 or 4q+3 where q is some integer​

Answer 1

$\huge{\blue{\bold{\underline{\underline{Answer :}}}}}$

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$\large{\red{\underline \bold{\tt{To \: Prove :-}}}}$

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• Every positive odd integer is of the form 4q + 1 or 4q + 3 where q is some integer

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$\underline{\bold{\texttt{By Euclid's Division Lemma }}}$

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If a and b are 2 positive integers, then

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➠ a = bq +r

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Where 0 ≤ r < b

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• Let "a" be a positive integer

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• b = 4

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Hence , a = 4q + r

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Where 0 ≤ r < 4

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"r" is an integer greater than or equal to 0 and less than 4

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Hence r can be either 0 , 1 , 2 or 3

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Case 1

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➠ Let r = 0

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➜ a = 4q + r

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➜ a = 4q

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➠ a = 2(4q)

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As it is divisible by 2 hence it is an even number

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Case 2

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➠ Let r = 1

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➜ a = 4q + r

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➠ a = 4q + 1

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As it is not divisible by 2 hence it is an odd number

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Case 3

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➠ Let r = 2

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➜ a = 4q + r

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➜ a = 4q + 2

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➠ a = 2(2q + 1)

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As it is divisible by 2 hence it is an even number

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Case 4

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➠ Let r = 3

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➜ a = 4q + r

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➠ a = 4q + 3

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As it is not divisible by 2 hence it is an odd number

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We got odd numbers in Case 2 , Case 4 and from there we got that odd integer is of the form 4q + 1 or 4q + 3 where q is some integer

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Hence proved

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