Find the remainder when 1^2019 + 2^2019 + 3^2019 + ......2020^2019 is divided by 2019.
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39
Answer:
1
Step-by-step explanation:
Given :
To find the remainder,
when,
⇒ is divided by 2019,.
Solution :
We know that,
for any two numbers with same odd powers, their sum is divisible by them,.
i.e.,
Let a & b be two integers,.
then,. iff n is odd , is divisible by a + b,.
Let a + b = 2019 , n = 2019,.
⇒ ⇒ is divisible by 2019,.
⇒ ⇒ is divisible by 2019,.
Then, by dividing the numbers into suitable pairs,
⇒
⇒ is divisible by 2019,.
also is divisible by 2019,.
⇒ Hence,. The last 2020²⁰¹⁹ can be written as,.
(by adding and subtracting (-1)²⁰¹⁹)
⇒
As, (-1)²⁰¹⁹ = -1
⇒ is divisible by 2019,.
(-1)²⁰¹⁹ = -1 ,
There - (-1)²⁰¹⁹ = -(-1)= 1 ,. is remaining (extra)
hence, the remainder is 1,.
sivaprasath:
nice Question, very tricky,.
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