Numbers-9 (XAT-2011) (5-Marks)

Let a_n = 111111…1, where 1 occurs n number of times. Then,

(i) a₇₄₁ is not a prime

(ii) a₅₃₄ is not a prime

(iii) a₁₂₃ is not a prime

(iv) a₇₇ 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 Click on "Read more" below:

Solution:

a₇₄₁ 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

a₅₃₄ has 534 number of 1’s => even number of 1’s

=> 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

a₁₂₃ 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

Being (i),(ii),(iii) options correct we can eliminate the answer options (1),(2),(3) and safely assume that a₇₇ is also not a prime.

Answer (4)