Question

If N = 1223334444…….and is a 100-digit number, find the remainder when N is divided by 16.
A. 15
B. 13
C. 11
D. 9

Answer 1

Hint: In order to solve this question, we should know the given number N’s each time its value is solved roughly. We can find out the last digits are 2121. Then using the divisibility rule of 16 in which we should know that if the thousands digits form a number that is divisible by 16. A number is also divisible by if the thousandth digit is odd (like in this case) and the number formed by the last three digits plus 8 is divisible by 16. So at last when we divide this number by 16 we will find the required remainder .

Complete step by step answer:
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 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.

So, the correct answer is “Option D”.

Note: In this question one can make a mistake about finding the digits in the last place. Also, one can be at fault when finding when using the rule for divisibility of 16. One must remember the basic operations as one can make a mistake there. By doing these basics one should be able to solve the question.