101,10101,1010101,101010101,10101010101,.........

the series goes on like this.

here only 101 is prime,all others are not.how u prove it.

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## Thursday, February 18, 2010

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Prime or not?

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101,10101,1010101,101010101,10101010101,.........

the series goes on like this.

here only 101 is prime,all others are not.how u prove it.

Posted in: Puzzles

## 1 Answers:

Solution :

Though Solution is too long, it works.

101 = 100+1

10101 = (100+1)*100+1 etc...

=> ((x+1)*x+1)*x+1 ....

= x^n + x^(n-1) + x^(n-2) .....

= (x^(n+1) -1) / x-1

=> (100^(n+1)-1) / 99

=> 9999/99(n=1) & 999999/99(n=2) & 99999999/99

=> 1111/11 , 111111/11 , 11111111/11.....

101*11 = 1111

1001*111 = 11111

10001*1111 = 11111111

& when odd number of 1's 100...1 is divisible by 11

& when even number of 1's occur 100...1 is indivisible by 11 but 111... is.

so one of them is divisible by 11 taking care of the dinominator in 1111/11 , 111111/11 , 11111111/11.....

Thus there remains, two terms in the numerator, making it non prime.....

The above four lines does'nt apply to 101*11 as 11/11 is 1 & there is only one term in the numerator for the first case.

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