## Sunday, 30 October 2011

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

Let an = 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
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 1st digit = sum of alternate digits starting with 2nd 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.