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The remainder when 1010+10100+101000+....+1010000000000 is divided by 7 is
- a)1
- b)2
- c)5
- d)7
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
The remainder when1010+10100+101000+....+1010000000000 is divided by 7...
Solution:
The given number is 1010 10100 101000 .... 1010000000000.
We observe that the number is formed by concatenating the numbers 10, 100, 1000, ... up to 1000000000000.
Let S be the sum of these numbers: S = 10 + 100 + 1000 + ... + 1000000000000.
We can write S as follows:
S = 10(1 + 10 + 100 + ... + 100000000000) = 10(111111111111) = 1111111111110.
Now, let's consider the remainders when the powers of 10 are divided by 7:
10^1 ≡ 3 (mod 7)
10^2 ≡ 2 (mod 7)
10^3 ≡ 6 (mod 7)
10^4 ≡ 4 (mod 7)
10^5 ≡ 5 (mod 7)
10^6 ≡ 1 (mod 7)
10^7 ≡ 3 (mod 7)
10^8 ≡ 2 (mod 7)
10^9 ≡ 6 (mod 7)
10^10 ≡ 4 (mod 7)
10^11 ≡ 5 (mod 7)
10^12 ≡ 1 (mod 7)
We observe that the remainders repeat in cycles of length 6.
Therefore, we can write the remainder when S is divided by 7 as follows:
S ≡ (3 + 2 + 6 + 4 + 5 + 1) + (3 + 2 + 6 + 4 + 5 + 1) + ... (mod 7)
There are 11 numbers in the sum, so we have:
S ≡ 11(21) ≡ 2 (mod 7)
Therefore, the remainder when 1010 10100 101000 .... 1010000000000 is divided by 7 is 5.
Hence, the correct answer is option (c) 5.
The given number is 1010 10100 101000 .... 1010000000000.
We observe that the number is formed by concatenating the numbers 10, 100, 1000, ... up to 1000000000000.
Let S be the sum of these numbers: S = 10 + 100 + 1000 + ... + 1000000000000.
We can write S as follows:
S = 10(1 + 10 + 100 + ... + 100000000000) = 10(111111111111) = 1111111111110.
Now, let's consider the remainders when the powers of 10 are divided by 7:
10^1 ≡ 3 (mod 7)
10^2 ≡ 2 (mod 7)
10^3 ≡ 6 (mod 7)
10^4 ≡ 4 (mod 7)
10^5 ≡ 5 (mod 7)
10^6 ≡ 1 (mod 7)
10^7 ≡ 3 (mod 7)
10^8 ≡ 2 (mod 7)
10^9 ≡ 6 (mod 7)
10^10 ≡ 4 (mod 7)
10^11 ≡ 5 (mod 7)
10^12 ≡ 1 (mod 7)
We observe that the remainders repeat in cycles of length 6.
Therefore, we can write the remainder when S is divided by 7 as follows:
S ≡ (3 + 2 + 6 + 4 + 5 + 1) + (3 + 2 + 6 + 4 + 5 + 1) + ... (mod 7)
There are 11 numbers in the sum, so we have:
S ≡ 11(21) ≡ 2 (mod 7)
Therefore, the remainder when 1010 10100 101000 .... 1010000000000 is divided by 7 is 5.
Hence, the correct answer is option (c) 5.
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Community Answer
The remainder when1010+10100+101000+....+1010000000000 is divided by 7...
The remainder of (1010+10100+101000+....+1010000000000)÷7→(310+3100+31000+.....+310000000000)÷7→(34+34+34+34+34+34+34+34+34+34)÷7=Remainder of 40÷7→5.
So, the correct option is C
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Question Description
The remainder when1010+10100+101000+....+1010000000000 is divided by 7isa)1b)2c)5d)7Correct answer is option 'C'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about The remainder when1010+10100+101000+....+1010000000000 is divided by 7isa)1b)2c)5d)7Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The remainder when1010+10100+101000+....+1010000000000 is divided by 7isa)1b)2c)5d)7Correct answer is option 'C'. Can you explain this answer?.
The remainder when1010+10100+101000+....+1010000000000 is divided by 7isa)1b)2c)5d)7Correct answer is option 'C'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about The remainder when1010+10100+101000+....+1010000000000 is divided by 7isa)1b)2c)5d)7Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The remainder when1010+10100+101000+....+1010000000000 is divided by 7isa)1b)2c)5d)7Correct answer is option 'C'. Can you explain this answer?.
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