whats the remainder ?

Question

N=121212..........upto 300 digits .

whats the remainder when N is divided by 999?

(1)12

(2)121

(3)216

(4)666

How to solve this type of problem ?

Answer 1

The answer is 666.
121212 ≡ 333 (mod 999)
1000001 ≡ 2 (mod 999)

so, 121212 * 1000001 ≡ 333*2 ≡ 666 (mod 999)
or 121212121212 ≡ 666 (mod 999)
& 121212121212*1000000 + 121212 ≡ 666*1000 + 333 ≡ 666333 ≡ 0 (mod 999)
so 121212121212121212 ≡ 0 (mod 999)

So the remainder cycles 333, 666, 0 after including 6 digits, 12 digits, 18 digits, etc., of N. Since N is 50 groups of 6 digits, and
50 ≡ 2 mod 3, N ≡ 666 mod 999

Answer 2

You could use the method of induction. That is, you take a small number, in your example , say 121212, which obeys the pattern and is larger than the divisor. This gives a remainder of 333. Try the next number, i.e. 12121212. This gives a remainder of 345. Try the next number, 1212121212. which gives you 546. Next number 121212121212 gives you 666. The next value 12121212121212 This leaves behind 678. Next 1212121212121212 leaves behind 879. The next value 121212121212121212 is perfectly divisible (leaves 0). Next 12121212121212121212 leaves just 12;
1212121212121212121212 leaves 213;
121212121212121212121212 again leaves 333 !! This cycle recurs. If you observe, when there are --
2 digits/ 20 digits / 38 digits the remainder is 12;
4 digits/ 22 digits / 40 digits the remainder is 213;
6 digits/ 24 digits / 42 digits the remainder is 333;
8 digits/ 26 digits / 44 digits the remainder is 345;
10 digits/ 28 digits / 46 digits the remainder is 546;
12 digits/ 30 digits / 48 digits the remainder is 666;
14 digits/ 32 digits / 50 digits the remainder is 678;
16 digits/ 34 digits / 52 digits the remainder is 879;
18 digits/ 36 digits / 54 digits the remainder is 0;

i.e., for every 18 digits the remainder returns to 0.
Hence extending this principle, if you have 300 digits, then highest number less than 300 AND divisible by 18 is 288. If you group off the digits in groups of 18, you will be left with 12 digits in the last group. So, your remainder should be - 666.

Answer 3

Let's begin with 12, and see how the remainder is changed each time we multiply by 100 and add 12.
(150 times)

1. 12 - the remainder is 12
2. 12 * 100 = 1200 = 999 + 201
1212 = 999 + 201 + 12 = 999 + 213
Now, we multiply the remainder by 100 and add 12
3. 213*100 + 12 = 21,312 = 21 * 999 + 333
4. 333*100 + 12 = 33,312 = 33 * 999 + 345
5. 345*100 + 12 = 34,512 = 34 * 999 + 546
6. 546*100 + 12 = 54,612 = 54 * 999 + 666
7. 666*100 + 12 = 2*(333*100 + 12) - 12 =
= 2*(33*999 + 345) - 12 = 66*999 + 690-12 =
= 66*999 + 678
8. 678*100+12 = 67,812 = 67*999 + 879
9. 879*100+12 = 87,912 = 88

Now, you see that every 9 times you add "12" the remainder becomes 0.

Let's divide 150 by 9 and get 16 with the remainder 6.

At the 6th step the remainder is 666 (answer 4).

Answer 4

jadageesh's answer is good and sensible.

Answer 5

I got 121.33333333..........