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.