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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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New post 19 Jul 2019, 21:31
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

Sol:
HCF is the greatest common factor for all the numbers.

Considering statement (1) alone:
X = 32 = 2^5
Y = 81 = 3^4
As the two numbers only have 1 as the common factor, the HCF of the three numbers X, Y, and Z is 1.
SUFFICIENT

Considering statement (2) alone:
Y = 81 = 3^4
Z = 121 = 11^2
As the two numbers only have 1 as the common factor, the HCF of the three numbers X, Y, and Z is 1.
SUFFICIENT

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

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New post 19 Jul 2019, 21:45
1
A tricky question. At first glance answer is C, but it is a trap answer
Let's prime factorize to find GCF:
Statement 1. X=\(2^5\), Y=\(3^4\)
The only factor they share is 1. No matter what number is Z, GCF will have to be 1 because only 1 is common between X and Y. A is Sufficient
Statement 2. Y=\(3^4\), Z=\(11^1\). Again, Y and Z share only 1 as their common factor. Whatever number is X, it will not affect GCF of X, Y, and Z because only 1 is common between Y and Z. Sufficient
Answer is D
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Re: What is the highest common factor of three positive integers X, Y, and  [#permalink]

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New post 19 Jul 2019, 21:52
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What is the highest common factor of three positive integers X, Y, and Z?

(1) X = 32, Y = 81

\(X = 2^5 y = 3^4.\) Hence common factor of X and Y is 1. Given that Z is a positive integer, 1 will be common between X, Y and Z and no other factor can be common between X, Y and Z. Hence, it is sufficient.

(2) Y = 81, Z = 121
\(Y = 3^4 z = 11^2.\) Here common factor between Y and Z is 1. Given that X is a positive integer, 1 will be common between X, Y and Z and no other factor can be common between X, Y and Z. Hence, it is sufficient.

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

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New post 19 Jul 2019, 22:06
1
The answer should be D. As each statement alone is sufficient.

In both the statements the 2 nos given are co-prime (prime to each other). So the highest common factor between the two of them is 1. Whatever the third no be in both the cases, the HCF of 3 numbers has to be 1 as the HCF of the given 2 number is 1.

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

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New post 19 Jul 2019, 22:13
1
1
We need to find GCD of 3 positive integers

(1) X = 32, Y = 81...> GCD of X and Y is:

X= 2*2*2*2*2 Y= 3*3*3*3

Greatest Common factor =1
Now, GCD of all three numbers will always be 1 irrespective of 3rd number. So, this is sufficient

(2) Y = 81, Z = 121

Y= 3*3*3*3 Z= 11*11

Greatest common factor =1

With same reasoning, this equation is also sufficient.

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

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New post 19 Jul 2019, 22:24
1
Considering (1) X = 32, Y = 81
Since 32=2^5*1
and 81= 3^4*1
Since X and Y have only 1 as the common factor we don't need Z to decide HCF of X,Y,Z. I will be 1.
Hence sufficient

(2) Y = 81, Z = 121

81=3^4*1
and 121=11^2*1
So as in Statement (1) 1 will be be only common factor and hence the HCF of X,Y,Z.
Hence sufficient

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

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New post 19 Jul 2019, 22:26
What is the highest common factor of three positive integers X, Y, and Z?

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

#1
x=2^5 and y=3 ^4
z value not know insufficient
#2
y=3^4 and z=11^2
x value not know insufficient
from 1 & 2
x=2^5,y=3^4 and z=11^2
HCF of x,y,z sufficient
IMO C
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Re: What is the highest common factor of three positive integers X, Y, and  [#permalink]

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New post 19 Jul 2019, 22:59
HCF of three given numbers can be find out by-
1) calculating HCF of any two numbers. and then
2) calculating HCF of the other two numbers with the HCF obtained in step 1.

consider x=3,y=6,z=12

A=HCF(x,y)=3
B=HCF(y,z)=6

HCF(A,B)=3

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

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New post 19 Jul 2019, 23:21
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What is the highest common factor of three positive integers X, Y, and Z?

So we need all the three values to answer the question

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

Combined we get all the values, therefore sufficient.
Answer is D
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Re: What is the highest common factor of three positive integers X, Y, and  [#permalink]

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New post 19 Jul 2019, 23:47
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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

St. 1 : X=32, Y=81 , no info for Z but since X= 32 can be written as 2^5 & Y=81 can be written as 3^4 and both do not share any common factor, whatever be the value for Z the highest common factor for all three variable would be 1
Hence Sufficient


St. 2 : Y=81 and Z= 121, no info for X but since Y=81 can be written as 3^4 & Z= 121 can be written as 11^2 and and both do not share any common factor, whatever be the value for X the highest common factor for all three variable would be 1
Hence Sufficient

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

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New post 19 Jul 2019, 23:54
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What is the HCF of three positive integers X,Y, and Z?

1. X=32, Y=81
Factors of X=32 {1, 2, 4, 8, 16, and 32}
Factors of Y=81 {1, 3, 9, 27, and 81}
It can be seen that the only common factor between 32 and 81 is 1. And we know every positive integer has 1 as factor. Irrespective of the value of Z, the only factor it will have in common with 32 and 81 will still be 1. Hence Statement 1 on its own sufficient.

2. Y=81, Z=121
Factors of Y=81 {1, 3, 9, 27, and 81}
Factors of Z=121 {1, 11, and 121}
HCF of Y and Z is 1. We know X is a positive integer which will have 1 as part of its factors. We can therefore determine that the HCF of X, Y, and Z is 1. Hence Statement 2 is also sufficient on its own.

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

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New post 20 Jul 2019, 00:07
What is the highest common factor of three positive integers X, Y, and Z?

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

Stmts 1 and 2 by themselves are clearly not sufficient because they only provide values for 2 of the 3 integers.

The two statements together we have, X=32, Y=81 and Z=121 with prime factorization of X = 2^5, Y=3^4 and Z=11^2 and sufficient to say the highest common factor will be 1.

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

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New post 20 Jul 2019, 00:10
1
Given,

X, Y, Z are positive integers.

To find,

Highest Common Factor of the 3 positive integers X, Y, Z.

Let us check the options now,

Option 1: X = 32, Y = 81

X = 2^5
Y = 3^4

=> Their HCF is 1.

Hence the HCF of X, Y, Z will be 1.

Option 1 is sufficient.

Option 2: Y = 81, Z = 121

Y = 3^4
Z = 11^2

=> Their HCF is 1.

Hence the HCF of X, Y, Z will be 1.

Option 2 is sufficient.

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

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New post 20 Jul 2019, 00:28
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What is the highest common factor of three positive integers X, Y, and Z?

(1) X = 32, Y = 81
X = 2^5 & Y = 3^4. Here, HCF of X and Y is "1". It does not matter what value 'Z' have, HCF for X, Y & Z will also be "1".

So, first can answer the question independently.


(2) Y = 81, Z = 121
Y = 3^4 & Z = 11^2. Here, HCF of Y and Z is "1". It does not matter what value 'X' have, HCF for X, Y & Z will also be "1".

So, second can answer the question independently.


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

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New post 20 Jul 2019, 00:28
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What is the highest common factor of three positive integers X, Y, and Z?

(1) X = 32, Y = 81, common factor is 1 only. X=2^5 and Y=3^4. Z can have both 2 and 3 but a HCF between three numbers is 1.
(2) Y = 81, Z = 121, common factor is 1 only. Y=3^4 and Z=11^2, X can have both 3 and 11, but a HCF between three numbers is 1.
Answer D
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Re: What is the highest common factor of three positive integers X, Y, and  [#permalink]

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New post 20 Jul 2019, 00:36
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highest common factor of three positive integers X, Y, and Z?


STATEMENT (1) X = 32, Y = 81
X = \(2^5\) Y = \(3^4\) x and y are co primes they have no factor in common except 1

any positive integer value of Z will not have a common factor with (X,Y and Z) except 1
suppose Z =6
Z have a common factor with X and Y (individually)
but X,Y and z combined doesnt have any common factor except 1

so X ,y and Z have highest common factor 1
SO SUFFICIENT

STATEMENT (2) Y = 81, Z = 121
Y and Z are co prime they have common factor 1
any positive integer value of X will not have a common factor with (X,Y and Z) except 1
so X,Y and Z highest common factor = 1
SUFFICIENT

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

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New post Updated on: 20 Jul 2019, 17:16
Question: What is the highest common factor of three positive integers X, Y, and Z?

(1) X = 32, Y = 81

\[\text{Prime Factor Calculation of 32}\\
\begin{pmatrix*}
&&32&& \\
&\swarrow&|&\searrow&\\
2&&\downarrow&&2\\
&&8&&\\
&\swarrow&&\searrow&\\
2^2&&&&2\\
&&&&
\end{pmatrix*}
\rightarrow \text{Prime Factors of 32 are } 2,2,2,2,2 = 2^5\\
\text{ } \\
\text{ } \\
\text{Prime Factor Calculation of 81}\\
\begin{pmatrix*}
&&81&& \\
&\swarrow&|&\searrow&\\
3&&\downarrow&&3\\
&&9&&\\
&\swarrow&&\searrow&\\
3&&&&3\\
&&&&
\end{pmatrix*}
\rightarrow \text{Prime Factors of 81 are } 3,3,3,3 = 3^4\\
\text{ } \\
\text{No prime factors are shared between the number 32 and 81. Therefore the only shared factor is 1. } \\
\]

(2) Y = 81, Z = 121

\[\text{Prime Factor Calculation of 81}\\
\begin{pmatrix*}
&&81&& \\
&\swarrow&|&\searrow&\\
3&&\downarrow&&3\\
&&9&&\\
&\swarrow&&\searrow&\\
3&&&&3\\
&&&&
\end{pmatrix*}
\rightarrow \text{Prime Factors of 81 are } 3,3,3,3 = 3^4\\
\text{ } \\
\text{ } \\
\text{Prime Factor Calculation of 121}\\
\begin{pmatrix*}
&&121&& \\
&\swarrow&&\searrow&\\
11&&&&11\\
&&&&
\end{pmatrix*}
\rightarrow \text{Prime Factors of 121 are } 11,11 = 11^2\\
\text{ } \\
\text{No prime factors are shared between the number 121 and 81. Therefore the only shared factor is 1. } \\
\]

Correct Answer: D \(\implies\) The greatest common factor of ALL three integers is 1.
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Originally posted by duchessjs on 20 Jul 2019, 00:51.
Last edited by duchessjs on 20 Jul 2019, 17:16, 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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New post 20 Jul 2019, 03:12
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(i) x = 32 --> 2^5, y = 91 --> 3^4, the common factor of x & y is 1, and for any value (odd or even) of z, 1 is one factor. So, the common factor of x, y, and z is 1, which happens to be the highest common factor, sufficient
Example: if z = 2, then we have 2^5, 3^4, and 2^1, the highest common factor is 1.
if z = 12 = 2^2*3, then we have 2^5, 3^4, and 2^2*3, the highest common factor is 1

(ii) y = 91 --> 3^4, z = 121 --> 11^2, 1 is the common factor of y and z. So, for any value of x, 1 is one common factor, and highest common factor of x, y, and z is 1, sufficient

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

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New post 20 Jul 2019, 03:22
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Statement 1:
\(X = 32 = 2^5\)
\(Y = 81 = 3^4\)
As there are no common factors between X and Y other than 1, HCF of X and Y = 1
Hence, even if Z has a common factor (other than 1) with either X or Y or both, the HCF of X, Y and Z will be 1.
Sufficient.

Statement 2:
\(Y = 81 = 3^4\)
\(Z = 121 = 11^2\)
As there are no common factors between Y and Z other than 1, HCF of Y and Z = 1.
Hence, even if X has a common factor (other than 1) with either Y or Z or both, the HCF of X, Y and Z will be 1.
Sufficient.

Option (D)
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Re: What is the highest common factor of three positive integers X, Y, and   [#permalink] 20 Jul 2019, 03:22

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