deep319933 wrote: ↑Wed Jun 24, 2020 12:31 am
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.
Solution:
Since the digits are the same, the sum of the digits of x is n + n + n + n = 4n, which is always even.
The product of the digits of x is n^4, which could be even or odd. For example, if x = 1111, the product of the digits is 1, which is odd. However, if x = 2222, the product of the digits is 16, which is even.
It’s true that x is not divisible by 12. In order to be divisible by 12, x has to be a multiple of both 3 and 4. Therefore, x could only be 3333, 6666 or 9999 if it’s a multiple of 3. However, none of these three numbers is divisible by 4 since the last two digits of any of these numbers is not divisible by 4. Alternatively, one could also list such numbers that are divisible by 4. The only possibilities are 4444 and 8888, none of which is divisible by 3.
Answer: D
