TITLE: Find the $44{th}$ digit of a $80$ digit number if number is divisible by $13$
QUESTION [8 upvotes]: $N$ is an $80-$digit positive integer (in the decimal scale). All digits except the $44th$ digit (from the left) are $2$. If N is divisible by $13,$ find the $44{th}$ digit ?
P.S: This isn't a homework question. Found it on a website and the explanation for the answer was very vague.
Link: http://iim-cat-questions-answers.2iim.com/quant/number-system/other/number-system_8.shtml

REPLY [3 votes]: 222222 is divisible by 13.
Therefore 222...(42 digits) is divisible by 13. Since (7 * 6 = 42)
And 222...(36 digits) is divisible by 13. Since (6 * 6 = 36)
Therefore 42 + 36 = 78 digits
We have the 43rd and 44th digit in the middle, which have to be divisible by 13.
We know 43rd digit is also 2.
26 is the only possibility which is divisible by 13.
Therefore 44th digit is 6.