Math, asked by akshatkhullar, 1 year ago

prove that the square of any positive integer is of the form 5q,5q+1,5q+4 for same integer q

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Answered by navyabhayana
2
,hope this will help u
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Answered by Anonymous
0

Step-by-step explanation:

AnswEr :

Let us Consider that a & b are two positive integers.

\:\bold{\underline{\underline{\sf{\pink{By,\: Using\; Euclid's\; Division\: lemma}}}}} :-

\implies a = bm + r & 0 < r < b

\implies b = 5 So, r can be 0, 1, 2, 3 & 4

\rule{150}2

\implies a = 5m + r

If r = 0

\implies a = 5m

\:\:\:\;\:\;\:\;\dag\small\bold{\underline{\underline{\sf{\red{Squaring}}}}} -

\implies\sf (5m^2)

\implies\sf 25m^2

\small\sf{Taking\: Common\: 5}

\implies\sf 5(5m^2)

\sf{So, \; 5q \:\;\;\;\;\;\;\:\:\:\;[Here,\; q \:=\: 5m^2]}

\rule{150}2

If r = 1

\implies\sf 5m + 1

\:\:\:\;\:\;\:\;\dag\small\bold{\underline{\underline{\sf{\red{Squaring}}}}} -

\implies\sf (5m + 1)^2

\sf{Using\: Identity\: (a +b)^2\:=\;a^2 + b^2 + 2ab}

\implies\sf 25m^2 + 1 + 10m

\small\sf{Taking\: Common\: 5}

\implies\sf 5(5m^2 + 2m) +1

\sf{So, 5q + 1 \:\;\;\;\;\;\;\:\:\:\;[Here,\; q \:=\: 5m^2 + 2m]}

\rule{150}2

If r = 2

\implies\sf 5m + 2

\:\:\:\;\:\;\:\;\dag\small\bold{\underline{\underline{\sf{\red{Squaring}}}}} -

\implies\sf (5m +2)^2

\implies\sf 25m^2 + 4 + 20m

\small\sf{Taking\: Common\: 5}

\implies\sf 5(5m^2+4m) + 4

\sf{So, \; 5q + 4 \:\;\;\;\;\;\;\:\:\:\;[Here,\; q \:=\: 5m^2 + 4m]}

\rule{150}2

If r = 3

\implies\sf 5m + 3

\:\:\:\;\:\;\:\;\dag\small\bold{\underline{\underline{\sf{\red{Squaring}}}}} -

\implies\sf (5m + 3)^2

\implies\sf 25m^2 + 30m + 9

\implies\sf 25m^2 + 30m + 5 + 4

\small\sf{Taking\: Common\: 5}

\implies\sf 5(5m^2 +  6m + 1)+4

\sf{So, 5q + 4 \:\;\;\;\;\;\;\:\:\:\;[Here,\; q \:=\: 5m^2 + 6m + 1]}

\rule{150}2

If r = 4

\implies\sf 5m + 4

\:\:\:\;\:\;\:\;\dag\small\bold{\underline{\underline{\sf{\red{Squaring}}}}} -

\implies\sf (5m +4)^2

\implies\sf 25m^2 + 40m + 16

\implies\sf 25m^2 + 40m + 15 + 1

\implies\sf 5(5m^2 + 8m + 3) +1

\sf{So, 5q + 1 \:\;\;\;\;\;\;\:\:\:\;[Here,\; q \:=\: 5m^2 + 8m +3]}

Hence, Square of any positive integer is of the form 5q, 5q+1, 5q+4 for some integer q.

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