If N = 1223334444…….and is a 100-digit number, what will be the remainder when N is divided by 16 is ____

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Detailed Solution

In this question we can see that each digit is coming the number of times its value.
As 1 would appear 1 time, 2 would appear 2 times and so on
So, from 1 to 9 we will have 1+2+3+4+5+6+7+8+9=45 digits,
Then, again 10 times 10,11 times 11,12 times 12,13 times 13 and so on . And according to this , we will have 1 and 0 occupying 20 terms, 1 and 1 forming 11 will occupy 22 terms .
So in total we have 45+20+22=87 terms.
Now we are short of 13 terms to make 100 terms.
So , the next number which would start is 12 formed by 1 and 2 . 6 times 12 will appear i.e. occupying 12 terms.
Now we are short of just 1 term which will be occupied by ’ 1 ‘ (a digit from 12) .
So now we will have the number with last few digits as 1212121
So, the last 4 digits will be 2121.
Now for any number to be divisible by 16 the last 4 digits must be divisible.
So , on dividing 2121 by 16, we get the remainder as 9
Hence, the remainder is 9 when N is divided by 16.

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