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# What is the highest common factor of three positive integers X, Y, and

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What is the highest common factor of three positive integers X, Y, and  [#permalink]

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19 Jul 2019, 08:00
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What is the highest common factor of three positive integers X, Y, and Z?

(1) X = 32, Y = 81
(2) Y = 81, Z = 121

 This question was provided by Experts Global for the Game of Timers Competition

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What is the highest common factor of three positive integers X, Y, and  [#permalink]

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Updated on: 19 Jul 2019, 09:28
1
1
(1) X = 32, Y = 81
32 and 81 are co-primes which means they do not have any common factor other than 1. So whatever be the value of Z, the GCD of X,Y & Z is going to be 1

1 is sufficient

(2) Y = 81, Z = 121

Similarly, 81 and 121 have only 1 common factor. So whatever be the value of Z, the GCD of X,Y & Z is going to be 1

2 is sufficient

Originally posted by firas92 on 19 Jul 2019, 08:06.
Last edited by firas92 on 19 Jul 2019, 09:28, edited 2 times in total.
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What is the highest common factor of three positive integers X, Y, and  [#permalink]

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Updated on: 20 Jul 2019, 01:41
1
HCF of three numbers is the highest factor that can divide all the three numbers.

(1) X = 32, Y = 81
X = 32 = 2*2*2*2*2
Y = 81 = 3*3*3*3
So HCF (32, 81) = 1

No matter what is the third number is, there is no number higher than 1 that can divide all the three numbers.

Sufficient

(2) Y = 81, Z = 121
Y = 3*3*3*3
Z = 11*11
So HCF (81, 31) = 1

No matter what is the third number is, there is no number higher than 1 that can divide all the three numbers.

Sufficient

Originally posted by Sayon on 19 Jul 2019, 08:07.
Last edited by Sayon on 20 Jul 2019, 01:41, edited 2 times in total.
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Re: What is the highest common factor of three positive integers X, Y, and  [#permalink]

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19 Jul 2019, 08:08
1

between X,Y and Z,even if two has common factor >1 , and not common for third then still the highest would be 1.

from 1: HCF between X and Y is 1, Z can be any value still common factor needs to be 1--> suff
from 2: same reasoning as statement 1.--> suff

So D
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Re: What is the highest common factor of three positive integers X, Y, and  [#permalink]

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19 Jul 2019, 08:09
1
What is the highest common factor of three positive integers X, Y, and Z?

(1) X = 32, Y = 81
(2) Y = 81, Z = 121

Option 1: X = 32 and Y = 81.
X is power of 2. powers of 2 from 2 till 32 and 1 are the factors of X.
Y is a power of 4. Powers of 3 and 1 are factors of Y.
So no factor is common between X and Y except 1. Hence 1 will common factor across Z. Sufficient

Option 2: Y =81 and Z =121.
Same as above. Y is power of 3 and Z is square of 11. Hence no common factor of Y and Z except 1. - Sufficient

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Re: What is the highest common factor of three positive integers X, Y, and  [#permalink]

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19 Jul 2019, 08:15
1
What is the highest common factor of three positive integers X, Y, and Z?

(1) X = 32, Y = 81
(2) Y = 81, Z = 121

Question: What is the highest common factor of three positive integers X, Y, and Z?

(1) X = 32, Y = 81
X= 32 = 2^5
Y= 81= 3^4
Highest Common Factor (HCF) of X & Y = 1 since there is no common factor except 1
HCF for X, Y & Z = 1 since HCF of (X,Y,Z) =< HCF (X,Y)
HCF (X,Y,Z) =1
For example, HCF(2,5) =1 => HCF(2,5,Z) =1 => HCF (2,5,7) = 1
SUFFICIENT

(2) Y = 81, Z = 121
Y= 81 = 3^4
Z= 121 = 11^2
HCF of Y & Z = HCF (Y,Z)=1 since there is no common factor except 1
HCF(X,Y,Z) = 1 since there is no HCF smaller than 1
For example, HCF(2,5) =1 => HCF(2,5,Z) =1 => HCF (2,5,7) = 1
SUFFICIENT

IMO D
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Re: What is the highest common factor of three positive integers X, Y, and  [#permalink]

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19 Jul 2019, 08:21
1
To find HCF of three positive integers.

(1) x=2^5 and y=3^4. HCF of x and y is 1, hence we can conclude that HCF of x,y,z must be 1 only. Sufficient.

(2) y=3^4 and z=11^2, HCF of y and z is 1, hence we can conclude that HCF of x,y,z must be 1 only. Sufficient.

D is correct.
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Re: What is the highest common factor of three positive integers X, Y, and  [#permalink]

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19 Jul 2019, 08:22
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D, maybe way off, but

1) 32 = s^5 & 81 = 3^4, one is the only common factor between them. So even if you know Z the greatest common factor among the three positive integers is 1.

2) 81 = 3^4 and 121 = 11^2, same as 1, GCF between the two is 1, so among the three 1 is the GCF.
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Re: What is the highest common factor of three positive integers X, Y, and  [#permalink]

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19 Jul 2019, 08:22
1
(1) X = 32, Y = 81 - Sufficient, HCF between these two numbers (2^5 and 3^4) is 1. so irrespective of the value of Z the HCF will still remain 1 for all three numbers x,y,z (as HCF between 1 and any number will be 1)
(2) Y = 81, Z = 121 - Sufficient, HCF between these two numbers (3^4 and 11^2) is 1. so irrespective of the value of X the HCF will still remain 1 for all three numbers x,y,z (as HCF between 1 and any number will be 1)

IMO D
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What is the highest common factor of three positive integers X, Y, and  [#permalink]

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Updated on: 19 Jul 2019, 08:44
1
Quote:
What is the highest common factor of three positive integers X, Y, and Z?

This is the Data Sufficiency (DS) question, and we need to find out if the statements below are sufficient to find the highest common factor for X, Y and Z. Let us analyze each statement separately:

Statement 1:
(1) X = 32, Y = 81

By substituting these numbers via prime factors, we get $$X = 32 = 2^5$$ and $$Y = 81 = 3^4$$
Even though we do not possess any information about Z, we can notice that information above is sufficient to find the highest common factor as it is 1. Thus, when adding the third number to these two, independent of its value, the highest common factor will remain 1.
Sufficient

Statement 2:
(2) Y = 81, Z = 121

By substituting these numbers with prime factors, we get $$Y = 81 = 3^4$$ and $$Z = 121 = 11^2$$
Here again, even though no information is available about X, this statement is sufficient just as in the example above.
The highest common factor is 1.
Sufficient.

Both statements are sufficient.

Originally posted by RusskiyLev on 19 Jul 2019, 08:25.
Last edited by RusskiyLev on 19 Jul 2019, 08:44, edited 1 time in total.
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Re: What is the highest common factor of three positive integers X, Y, and  [#permalink]

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19 Jul 2019, 08:28
IMO C

Statement 1 -> We need to know the value of Z. --> Insufficient
Statement 2 -> We need to know the value of X. --> Insufficient

(1) + (2) --> we know X = 32, Y = 81 and Z = 121. With this, we can find the highest common factor. --> Sufficient
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Re: What is the highest common factor of three positive integers X, Y, and  [#permalink]

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19 Jul 2019, 08:28
What is the highest common factor of three positive integers X, Y, and Z?

Fairly easy question. To get the unique value we need to know the value of x, y , z.
Lets see the statements.

(1) X = 32, Y = 81
This is no value of Z , hence insufficient.

(2) Y = 81, Z = 121
This does not give us the value of X, hence insufficient.

Combining the 2 statements, we know the value of X, Y, AND Z.

Hence it would be sufficient. So the answer is C.
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Re: What is the highest common factor of three positive integers X, Y, and  [#permalink]

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19 Jul 2019, 08:29
1
D

Tricky question. To find the highest common factor of three positive integers X, Y and Z, we should normally need to have some idea about these integers, except when highest common factor between two integers is 1, in which case the factor cant get any smaller irrespective of what the value of third integer is.

St 1: X = 2^5, Y = 3^4 - highest common factor between X and Y is 1. So doesn't matter what value of Z is, the common factor cant get any lower and will remain 1.
St 2: Y = 3^4, Z = 11^2 - highest common factor between Y and Z is 1. So doesn't matter what value of X is, the common factor cant get any lower and will remain 1

They are sufficient individually.
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Re: What is the highest common factor of three positive integers X, Y, and  [#permalink]

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19 Jul 2019, 08:29
The highest common factor of X, Y, Z?
(1) X = 32, Y = 81
As we do not know about Z, then we cannot calculate the HCF of all three.
Insufficient.

(2) Y = 81, Z = 121
Again, we don`t know about X, then we cannot calculate HCF of all three.
Insufficient.

Combining 1 and 2:
We get unique values for X = 32 , Y = 81 , Z = 121
Thus, HCF:
X =32 => $$2^5$$
Y = 81 => $$3^4$$
Z = 121 => $$11^2$$
HCF = 1

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Re: What is the highest common factor of three positive integers X, Y, and  [#permalink]

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19 Jul 2019, 08:32
To find the HCF of X,Y and Z

Statement 1: X = 32, Y = 81
Since no value for Z was given
Therefore, statement 1 is Not Sufficient (BCE)

Statement 2: Y = 81, Z = 121
Since no value was given for X
Therefore, statement 2 is Not Sufficient (CE)

Combining statement 1 and 2
We have X = 32, Y = 81, Z = 121

Therefore, the HCF can be calculated from the values

Sufficient

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Re: What is the highest common factor of three positive integers X, Y, and  [#permalink]

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19 Jul 2019, 08:33
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IMO D.

The highest common factor of three positive integers X, Y, and Z.

Statement 1: X = 32, Y = 81, Highest common factor of X and Y will be 1. Hence, the highest common factor between 1 and Z would be 1, no matter what.
Hence, this statement is sufficient.

Statement 2: Y = 81, Z = 121 , Highest common factor of Y and Z will be 1. Hence, the highest common factor between 1 and X would be 1, no matter what.
Hence, this statement is sufficient.

Therefore, the answer is either of the two options are sufficient.
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Re: What is the highest common factor of three positive integers X, Y, and  [#permalink]

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19 Jul 2019, 08:36
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(1) X = 32, Y = 81 Highest common factor=1. Sufficient
(2) Y = 81, Z = 121 Highest common factor=1. Sufficient
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Re: What is the highest common factor of three positive integers X, Y, and  [#permalink]

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19 Jul 2019, 08:38
As per my knowledge to know hcf of any 3 three positive integers we need to know the nos. Hence I have marked C.

If there is any catch, please let me know.

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Re: What is the highest common factor of three positive integers X, Y, and  [#permalink]

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19 Jul 2019, 08:39
1
What is the highest common factor of three positive integers X, Y, and Z?

(1) X = 32, Y = 81---------->> both have only 1 in common, hence whatever is Z, 1 will be the highest common factor.
(2) Y = 81, Z = 121-------->> Same as statement 1 we have only 1 in common.

EACH ALONE IS SUFFICIENT, D is the answer
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Re: What is the highest common factor of three positive integers X, Y, and  [#permalink]

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19 Jul 2019, 08:39
1
What is the highest common factor of three positive integers X, Y, and Z?

(1) X = 32, Y = 81
(2) Y = 81, Z = 121

if any number have no common factor except they will not have with others also
(1) X = 32, Y = 81 : No common divisor as $$X = 2^{5}$$and $$y = 3^4$$ hence 1 will be the greatest factor for X, Y and X
(2) Y = 81, Z = 121: No common divisor as $$Z = 11^2$$ and $$y = 3^4$$hence 1 will be the greatest factor for X, Y and X
Both are self sufficient
D
Re: What is the highest common factor of three positive integers X, Y, and   [#permalink] 19 Jul 2019, 08:39

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