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X is a number formed by the first 100 digits of the number N which is formed by writing the first 100 natural numbers one after another as follows: 1234567891011. Find the remainder when X is divided by 16?
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X is a number formed by the first 100 digits of the number N which is ...
Solution:
Let's first write down the first few numbers of N:
N = 1234567891011121314151617181920212223...
Now, we need to find the first 100 digits of N. We can see that the first 9 digits of N are simply the numbers 1 through 9. The next 90 digits are the numbers 10 through 99, each with 2 digits. So, we can write:
N = 12345678910111213141516171819202122...9899
Now, we need to find the first 100 digits of X, which is just the first 100 digits of N. Let's group the digits of N into blocks of length 4:
N = 1234 5678 9101 1121 3141 5161 7181 9202 1222 ... 9899
Notice that each block of length 4 contains the digits of consecutive numbers. For example, the first block contains the digits of the numbers 1, 2, 3, and 4. The second block contains the digits of the numbers 5, 6, 7, and 8, and so on. So, we can write:
N = 1234 5678 9101 1121 3141 5161 7181 9202 1222 ... 9899
X = 1234 5678 9101 1121 3141 5161 7181 9202 1222 ... 9899
Finding the Remainder:
To find the remainder when X is divided by 16, we just need to find the remainder when the sum of the digits of X is divided by 16. Let's add up the digits in each block:
1 + 2 + 3 + 4 = 10
5 + 6 + 7 + 8 = 26
9 + 1 + 0 + 1 = 11
1 + 1 + 2 + 1 = 5
3 + 1 + 4 + 1 = 9
5 + 1 + 6 + 1 = 13
7 + 1 + 8 + 1 = 17
9 + 2 + 0 + 2 = 13
1 + 2 + 2 + 2 = 7
...
Now, we just need to add up all of these sums:
10 + 26 + 11 + 5 + 9 + 13 + 17 + 13 + 7 + ...
We can see that each sum is congruent to 3 mod 16, so their sum is also congruent to 3 mod 16. Therefore, the remainder when X is divided by 16 is 3.
Community Answer
X is a number formed by the first 100 digits of the number N which is ...
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X is a number formed by the first 100 digits of the number N which is formed by writing the first 100 natural numbers one after another as follows: 1234567891011. Find the remainder when X is divided by 16?
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X is a number formed by the first 100 digits of the number N which is formed by writing the first 100 natural numbers one after another as follows: 1234567891011. Find the remainder when X is divided by 16? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about X is a number formed by the first 100 digits of the number N which is formed by writing the first 100 natural numbers one after another as follows: 1234567891011. Find the remainder when X is divided by 16? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for X is a number formed by the first 100 digits of the number N which is formed by writing the first 100 natural numbers one after another as follows: 1234567891011. Find the remainder when X is divided by 16?.
X is a number formed by the first 100 digits of the number N which is formed by writing the first 100 natural numbers one after another as follows: 1234567891011. Find the remainder when X is divided by 16? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about X is a number formed by the first 100 digits of the number N which is formed by writing the first 100 natural numbers one after another as follows: 1234567891011. Find the remainder when X is divided by 16? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for X is a number formed by the first 100 digits of the number N which is formed by writing the first 100 natural numbers one after another as follows: 1234567891011. Find the remainder when X is divided by 16?.
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