suneja
If n is a positive integer, what is the remainder when 3^(8n+3)+2 is divided by 5?
A. 0
B. 1
C. 2
D. 3
E. 4



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him1618
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Split it up into 3^3.3^(8n) +2

him1618
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Split the second 3 as (52)..expand binomially

suneja
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didnt gt u

him1618
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binomial expansion aata hai?

suneja
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@him1618 if u tak n=1 ans is 4 i want does tis hold true for all n

him1618
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that isnt the way to do it though

suneja
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then

suneja
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how to go abt it

him1618
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like i said
remodel it as
27 (52)^(8n) +2
expand (52) part binomially

him1618
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when u expand it
ull get all terms with a 5 or some power of 5 in them
except fr the last one
so ure left with
27(5q  2) +2

waterineyes
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\(3^{8n}\) if 3 has powers which are divisible by 4, then Remainder it has is 1..
Here : 8n is divisible by 4..

waterineyes
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Similary, \(3^3\) will give you 7 as Unit place..

waterineyes
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So:
\[3^{8n} \cdot 3^3 + 2 \implies 1 \cdot 7 + 2 \implies 9\]

waterineyes
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So, what will you get as remainder when you divide 9 by 5 ??