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An integer X is a multiple of 8

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An integer X is a multiple of 8 [#permalink] New post 01 Oct 2018, 15:55
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46% (01:15) correct 53% (00:49) wrong based on 41 sessions
An integer X is a multiple of 8, 14 and 33. WHich of the following is a factor of X.

Indicate all possible values.

A. 16
B. 24
C. 77
D. 81
[Reveal] Spoiler: OA

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Re: An integer X is a multiple of 8 [#permalink] New post 12 Oct 2018, 23:22
why did u omit option A 16 is also correct along with option B and option C as 16 means 2*2*2*2 and the multiples are 8 14 33 which have factors 2*2*2*(2*7)*(3*11) or am i missing something
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Re: An integer X is a multiple of 8 [#permalink] New post 13 Oct 2018, 16:51
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snowbrood wrote:
why did u omit option A 16 is also correct along with option B and option C as 16 means 2*2*2*2 and the multiples are 8 14 33 which have factors 2*2*2*(2*7)*(3*11) or am i missing something


No 16 is not a factor.

We can rewrite the numbers in terms of factors:

\(8 = 2^3\)

\(14 = 2\times 7\)

\(33 = 3^11\)

So X must have = 2^3 \times 3 \times 7 \times 11.

\(16= 2^4\) This may not be a factor. For example, 1848 is a possible value of X but \(\frac{1848}{16}=115.5\)
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Re: An integer X is a multiple of 8 [#permalink] New post 30 Apr 2020, 23:01
sandy wrote:
snowbrood wrote:
why did u omit option A 16 is also correct along with option B and option C as 16 means 2*2*2*2 and the multiples are 8 14 33 which have factors 2*2*2*(2*7)*(3*11) or am i missing something


No 16 is not a factor.

We can rewrite the numbers in terms of factors:

\(8 = 2^3\)

\(14 = 2\times 7\)

\(33 = 3^11\)

So X must have = 2^3 \times 3 \times 7 \times 11.

\(16= 2^4\) This may not be a factor. For example, 1848 is a possible value of X but \(\frac{1848}{16}=115.5\)




But, 8*14*33 is 3696, which is a factor of 16. So why not option A?
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Re: An integer X is a multiple of 8 [#permalink] New post 30 Apr 2020, 23:18
Preetham wrote:
sandy wrote:
snowbrood wrote:
why did u omit option A 16 is also correct along with option B and option C as 16 means 2*2*2*2 and the multiples are 8 14 33 which have factors 2*2*2*(2*7)*(3*11) or am i missing something


No 16 is not a factor.

We can rewrite the numbers in terms of factors:

\(8 = 2^3\)

\(14 = 2\times 7\)

\(33 = 3^11\)

So X must have = 2^3 \times 3 \times 7 \times 11.


\(16= 2^4\) This may not be a factor. For example, 1848 is a possible value of X but \(\frac{1848}{16}=115.5\)




But, 8*14*33 is 3696, which is a factor of 16. So why not option A?


As shown above, 16 does not satisfy the condition all times, For instance, when divided by 3696, it gives us a factor, but in case of 1848 it doesn't. Similarly, the other two numbers (24,77) always satisfy the condition.
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Re: An integer X is a multiple of 8 [#permalink] New post 28 May 2020, 01:51
BUT HOW ARE WE SUPPOSED TO KNOW THAT 16 WILL NOT BE INCLUDED
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Re: An integer X is a multiple of 8 [#permalink] New post 28 May 2020, 02:10
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akkaur0310 wrote:
BUT HOW ARE WE SUPPOSED TO KNOW THAT 16 WILL NOT BE INCLUDED


Sir, this is a hard question due to the fact that it has a trap in it. We need to crosscheck each and every option before selecting the answer. In this case, if one does not crosscheck for other numbers which have same multiples (like 1848), then they've fallen for the trap.
You're not supposed to know everything, such questions check how aware you're of the fact that there are other multiples of the given number other than 3696.
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Re: An integer X is a multiple of 8 [#permalink] New post 28 May 2020, 03:12
Can anyone explain elaborately.
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Re: An integer X is a multiple of 8 [#permalink] New post 28 May 2020, 09:08
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sandy wrote:
An integer X is a multiple of 8, 14 and 33. Which of the following is a factor of X?

Indicate all possible values.

A. 16
B. 24
C. 77
D. 81


-----ASIDE---------------------
A lot of integer property questions can be solved using prime factorization.
For questions involving divisibility, divisors, factors and multiples, we can say:
If N is a multiple of k, then k is "hiding" within the prime factorization of N

Consider these examples:
24 is a multiple of 3 because 24 = (2)(2)(2)(3)
Likewise, 70 is a multiple of 5 because 70 = (2)(5)(7)
And 112 is a multiple of 8 because 112 = (2)(2)(2)(2)(7)
And 630 is a multiple of 15 because 630 = (2)(3)(3)(5)(7)
-----ONTO THE QUESTION!---------------------

Given: X is a multiple of 8
Since 8 = (2)(2)(2), we know that there are three 2's hiding in the prime factorization of X
So, X = (2)(2)(2)(?)(?)(?)....
Note: The additional (?)'s represents other possible prime numbers in the prime factorization of X.
For the moment, however, all we know for certain is that there are three 2's hiding in the prime factorization of X


Given: X is a multiple of 14
14 = (2)(7), we know that there is one 2 and one 7 hiding in the prime factorization of X
Since we already have a 2 in our prime factorization above, we need only add a 7 to our prime factorization.
We get: X = (2)(2)(2)(7)(?)(?)(?)....

Given: X is a multiple of 33
Since 33 = (3)(11), we know that there is one 3 and one 11 hiding in the prime factorization of X
So let's add them to our prime factorization.
We get: X = (2)(2)(2)(7)(3)(11)(?)(?)(?)....

So, X = (2)(2)(2)(3)(7)(11)(?)(?)(?)....
We can now see that X is a multiple of 8, 14, and 33

Now let's turn our attention to the answer choices:
A. 16
16 = (2)(2)(2)(2)
So, in order for 16 to be a factor, X must contain four 2's in its prime factorization
Since X does NOT contain four 2's in its prime factorization, 16 is NOT a factor of X

B. 24
24 = (2)(2)(2)(3)
Since X = (2)(2)(2)(3)(7)(11), we can see that 24 IS a factor of X

C. 77
77 = (7)(11)
Since X = (2)(2)(2)(3)(7)(11), we can see that 77 IS a factor of X

D. 81
81 = (3)(3)(3)(3)
So, in order for 81 to be a factor, X must contain four 3's in its prime factorization
Since X does NOT contain four 3's in its prime factorization, 81 is NOT a factor of X


Answer: B, C

Cheers,
Brent
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Re: An integer X is a multiple of 8 [#permalink] New post 07 Jul 2020, 01:12
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sukrut96 wrote:
akkaur0310 wrote:
BUT HOW ARE WE SUPPOSED TO KNOW THAT 16 WILL NOT BE INCLUDED


Sir, this is a hard question due to the fact that it has a trap in it. We need to crosscheck each and every option before selecting the answer. In this case, if one does not crosscheck for other numbers which have same multiples (like 1848), then they've fallen for the trap.
You're not supposed to know everything, such questions check how aware you're of the fact that there are other multiples of the given number other than 3696.



I see one simple way of doing this. just take the LCM of all the multiple and then check for factors
Re: An integer X is a multiple of 8   [#permalink] 07 Jul 2020, 01:12
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