Let a

_{n}= 111111…1, where 1 occurs n number of times. Then,
(i) a

_{741}is not a prime
(ii) a

_{534}is not a prime
(iii) a

_{123}is not a prime
(iv) a

_{77}is not a prime
(1) (i) is correct

(2) (i) and (ii) correct

(3) (ii) and (iii)
correct

(4) All of them are
correct

(5) None of them is
correct

For Answer
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__Solution:__**a**has 741 number of 1’s => sum of all digits = 7+4+1 = 12 being multiple of 3, it is not a prime => option (i) is correct

_{741}

**a**has 534 number of 1’s => even number of 1’s

_{534}
=> sum of alternate
digits starting with 1

^{st}digit = sum of alternate digits starting with 2^{nd}digit (rather we can say, sum of digits in all even places = sum of digits in all odd places)
=> it is a multiple of
11 => it is not a prime => option (ii) is also correct

**a**has 123 number of 1’s => sum of all digits = 1+2+3 = 6 being multiple of 3, it is not a prime => option (iii) is also correct

_{123}
Being (i),(ii),(iii) options
correct we can eliminate the answer options (1),(2),(3) and safely assume that

**a**is also not a prime._{77}**Answer (4)**
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