JMO - division

If P =1234567891011.............9989991000

find the Remainder when the first 1000 digits are divided by 16.



Answer:

number of one digit numbers = 9 - 1 + 1 = 9
number of  2 digit numbers = 99 - 10 + 1 = 90
total number of digits in the 2 digit numbers = 2 x 90 = 180

number of digits remaining for the 1000th digit = 1000 - (9 + 180) = 811

to establish the 3 digit number at the 1000th place we divide 811/3 = 270 (1/3)

therefore, the natural number where 1000th term will be located is  9+ 90 + 270 = 369

please note the fraction 1/3 above. It means the first number of a three digit number.

so, the 1000th term is 3693.

Now for divisibility test of 16, the last 4 numbers should be divisible by 16.

therefore, 3693/16 = 230(13/16)

SO, the Remainder is 13.