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x is a 4-digit positive integer whose digits are all the integer n. Wh

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x is a 4-digit positive integer whose digits are all the integer n. Wh  [#permalink]

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New post 19 Jun 2018, 23:45
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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

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x is a 4-digit positive integer whose digits are all the integer n. Wh  [#permalink]

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New post 19 Jun 2018, 23:55
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


The 4-digit positive number x = nnnn

i. The sum of digits is 4n. Any number(n) * Even number is always even irrespective
of whether n is odd or even. This statement is always TRUE
ii. The product of digits is n^4.
If n = 1(odd), product = odd | n = 2(even), product = even. Statement is not TRUE.
iii. 12 when prime-factorized gives 2^2 * 3. The number(x) must be divisible by 2 and 3.
Whatever the number, it can never be divisible by 12. This statement is TRUE

Therefore, Option D(i and iii only) must always be TRUE.
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Re: x is a 4-digit positive integer whose digits are all the integer n. Wh  [#permalink]

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New post 20 Jun 2018, 00:06
1
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
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Re: x is a 4-digit positive integer whose digits are all the integer n. Wh  [#permalink]

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New post 20 Jun 2018, 04:27
I am confused as how I is true.

If the number is 3332, 3+3+3+2 = 11. so how can i. be true?
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Re: x is a 4-digit positive integer whose digits are all the integer n. Wh  [#permalink]

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New post 21 Jun 2018, 04:57
vmelgargalan wrote:
I am confused as how I is true.

If the number is 3332, 3+3+3+2 = 11. so how can i. be true?
..


Given All 4 digits of the integer are same … Eg: 3333 …
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Re: x is a 4-digit positive integer whose digits are all the integer n. Wh  [#permalink]

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New post 24 Jun 2018, 17:56
1
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


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
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Re: x is a 4-digit positive integer whose digits are all the integer n. Wh  [#permalink]

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New post 22 Aug 2018, 08:04
GyMrAT wrote:
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.

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Re: x is a 4-digit positive integer whose digits are all the integer n. Wh &nbs [#permalink] 22 Aug 2018, 08:04
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