N is an 80 digit positive integer in the decimal scale All digits except the 44th digit from the lef...........

Quant (SSC) Number System

N is an 80-digit positive integer (in the decimal scale). All digits except the 44^th digit (from the left) are 2. If N is divisible by 13, find the 44^th digit?

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Explanation:

Any number of the form abcabc is a multiple of 1001.
1001 is 7 * 11 * 13.
So, any number of the form abcabc is a multiple of 13.
42 digits ......... 43 44 .... remaining 36 digits
222222 ..... 222222 2 a 222222............222222
Or we can write this 80 digit number as,
222222 ......2222220000000 ............000000 (It has 42 2's and 38 0's and it is divisible by 13)
+ 00000 .......00000 2 a 000000............000000 (43^rd digit is 2 and 44^th digit is a)
+ 00000 ..........000000 222222.............222222 (It has 36 2's in the end and it is divisible by 13)
We are left with a two digit number 2a. 26 is a multiple of 13, so the 44^th digit should be 6.