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A hundred and twenty digit number is formed by writing the first x natural numbers in front of each other as 12345678910111213… Find the remainder when this number is divided by 8.
- a)6
- b)7
- c)2
- d)0
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
A hundred and twenty digit number is formed by writing the first x nat...
The last 3 digits of the number would determine the remainder when it is divided by 8.
The number upto the 120th digit would be 1234567891011… 646.
646 divided by 8 gives us a remainder of 6.
The number upto the 120th digit would be 1234567891011… 646.
646 divided by 8 gives us a remainder of 6.
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A hundred and twenty digit number is formed by writing the first x nat...
1415161718192021... Find the value of x.
We can start by finding the number of digits used to write the first n natural numbers. This can be done using the formula for the sum of an arithmetic series:
S = n/2 (2a + (n-1)d)
where S is the sum of the first n natural numbers, a is the first term (which is 1), and d is the common difference (which is also 1). Simplifying this formula gives:
S = n/2 (n + 1)
Now we need to find the smallest value of n for which S is greater than or equal to 120. Setting S equal to 120 and solving for n gives:
120 = n/2 (n + 1)
240 = n^2 + n
n^2 + n - 240 = 0
Using the quadratic formula to solve for n gives:
n = (-1 ± sqrt(1 + 4*240))/2
n ≈ 15.56 or n ≈ -16.56
Since n must be a positive integer, the only possible value for n is 16. We can check that this is indeed the correct value by finding the number of digits used to write the first 16 natural numbers:
S = 16/2 (16 + 1) = 136
This is greater than 120, so we know that we need to use the first 15 natural numbers and part of the 16th natural number to form the hundred and twenty digit number. Specifically, we need to use the first 15 natural numbers in their entirety, and the first 5 digits of the 16th natural number (which is 16). Therefore, the hundred and twenty digit number is:
1234567891011121314151617
And the value of x is 15.
We can start by finding the number of digits used to write the first n natural numbers. This can be done using the formula for the sum of an arithmetic series:
S = n/2 (2a + (n-1)d)
where S is the sum of the first n natural numbers, a is the first term (which is 1), and d is the common difference (which is also 1). Simplifying this formula gives:
S = n/2 (n + 1)
Now we need to find the smallest value of n for which S is greater than or equal to 120. Setting S equal to 120 and solving for n gives:
120 = n/2 (n + 1)
240 = n^2 + n
n^2 + n - 240 = 0
Using the quadratic formula to solve for n gives:
n = (-1 ± sqrt(1 + 4*240))/2
n ≈ 15.56 or n ≈ -16.56
Since n must be a positive integer, the only possible value for n is 16. We can check that this is indeed the correct value by finding the number of digits used to write the first 16 natural numbers:
S = 16/2 (16 + 1) = 136
This is greater than 120, so we know that we need to use the first 15 natural numbers and part of the 16th natural number to form the hundred and twenty digit number. Specifically, we need to use the first 15 natural numbers in their entirety, and the first 5 digits of the 16th natural number (which is 16). Therefore, the hundred and twenty digit number is:
1234567891011121314151617
And the value of x is 15.
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