101,10101,1010101,101010101,10101010101,.........
the series goes on like this.
here only 101 is prime,all others are not.how u prove it.
101,10101,1010101,101010101,10101010101,.........
the series goes on like this.
here only 101 is prime,all others are not.how u prove it.
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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