1
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Question

# The remainder when 1+21+22+23............21999 is divided by 5 is:

A

0

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B

1

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C

2

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D

3

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Solution

## The correct option is A 0 The given sequence is Geometric progression with first term 1 and common ratio 2. 1+21+22+23............21999=22000−12−1 = 22000−1 22000−1=41000−1=(5−1)1000−1 (5−1)1000−1 =[1000C0(5)1000−1000C1(5)999+1000C2(5)998........−1000C999(5)+1000C1000]−1 (5−1)1000−1 =[1000C0(5)1000−1000C1(5)999+1000C2(5)998........−1000C999(5)+1]−1 (5−1)1000−1 =1000C0(5)1000−1000C1(5)999+1000C2(5)998........−1000C999(5) (5−1)1000−1 = A multiple of 5 ⇒ The remainder will be zero.

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