Bunuel wrote:

x is a 4-digit positive integer whose digits are all the integer n. Which of the following must be true?

i. The sum of the digits of x is even.

ii. The product of the digits of x is even.

iii. It is not divisible by 12.

A. i only

B. iii only

C. i and ii only

D. i and iii only

E. ii and iii only

If i understand the question correctly then we have a 4 digit number whose digits are all same & are equal to integer n.

Hence x = 1000n + 100n + 10n + n

& n = 1,2,3....9

i. The sum of the digits of x is even.

if n - odd/even, then we have sum of digits of x = 4n = even

hence this is true

ii. The product of the digits of x is even.

if n - odd, product of digits of x = n^4 = odd^4 = odd

if n - even, product of digits of x = n^4 = even^4 = even

Hence this can't be proved true.

iii. It is not divisible by 12.

x is not divisible by 12 = 3 * 2^2

x = 1000n + 100n + 10n + n = 1111n = (11 * 101)n

for x to be divisible by 12, n has to be a multiple of 12, which is not possible as integer n = 1,2...9

Hence n is not divisible by 12, statement (iii) is true.

Answer D.

Thanks,

GyM

Hey, can you please explain why n should be a multiple of 12 to make x divisible by 12? I partially understand the logic that n should be 1,2,3...9, and any formation can not be divisible by 12, but what's the full logic? guess I am missing a concept here, please enlighten me.