x is a 4digit 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.
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So the number x is of the format nnnn
1) Sum of the digits = 4n (which is divisible by 4, hence even)
2) Product of the digits = \(n^4\)
If n=even, product will be even. If n=odd, product will be odd. So, we can't be sure of this one.
3) To be divisible by 12, the number must be divisible by both 3 and 4.
To be divisible by 4, last 2 digits must be divisible by 4. Since the digits are same and number is positive, only two possibilities here > 4444 and 8888
Both of these numbers aren't divisible by 3
Hence, x is not divisible by 12
To summarise, only 1) and 3) must be true!
Drop a thanks if this helps!
1) Sum of the digits = 4n (which is divisible by 4, hence even)
2) Product of the digits = \(n^4\)
If n=even, product will be even. If n=odd, product will be odd. So, we can't be sure of this one.
3) To be divisible by 12, the number must be divisible by both 3 and 4.
To be divisible by 4, last 2 digits must be divisible by 4. Since the digits are same and number is positive, only two possibilities here > 4444 and 8888
Both of these numbers aren't divisible by 3
Hence, x is not divisible by 12
To summarise, only 1) and 3) must be true!
Drop a thanks if this helps!
1) Sum of the digits = 4n
Hence it's divisible by 4.
2) Product of the digits = n^4
For even or odd, it depends whether n is even or odd
3) To be divisible by 12, the number must be divisible by both 3 and 4.
Only possible numbers divisible by 4 will be 4444 and 8888, and none of them is divisible by 3
So, x is not divisible by 12
Hence it's divisible by 4.
2) Product of the digits = n^4
For even or odd, it depends whether n is even or odd
3) To be divisible by 12, the number must be divisible by both 3 and 4.
Only possible numbers divisible by 4 will be 4444 and 8888, and none of them is divisible by 3
So, x is not divisible by 12
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Solution:deep319933 wrote: ↑Wed Jun 24, 2020 12:31 amx is a 4digit 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.
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