3i think multipling and dividing the no by 9 and then we have to use binomial
prove 1111111111111111................(91 digits) is not a prime no.
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11 Answers
3
1111.......1 consists of 91 digits
Therefore 111...1= 1090+1089+1088+......+10+1
=1091-1/10-1
=[(107)13-1/107-1] *[107-1/10-1]
=(1+107+1014+....+1084)*(1+10+102+...+106)
Since it can be broken into factors,so it is not prime
66just for fun: the prime factorization turns out to be
11111 ...............(91 times)
= 53 X 79 X 239 X 547 X 4649 X 14197 X 17837 X 4262077 X 265371653 X 43442141653 X 316877365766624209 X 110742186470530054291318013
341\underbrace{\underbrace{11...111}_{\text{seven} \ 1's}\underbrace{11...111}_{\text{seven} \ 1's}...\underbrace{11...111}_{\text{seven} \ 1's}}_{13 \ \text{such units}
and is hence divisible by 1111111
62To add to this one.. (not for the experts ;P)
is this number given above a perfect square? Why?
11This no. given above is of the form 4n-1...........and 4n-1 cannot be a perfect square.
A no. ending with the digit 1 can be a perfect square iff " the no. formed by its preceding digits is divisible by 4 ".