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

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

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.

Answer: D

IMO : D

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) GCF of 32 and 81 is = 1 because there is no comman factor other than 1.
we know that if 1 is the gcf of x and y then for x y and z the gcf will be 1 as well.

So sufficient.

2)GCF of 81 and 121 is =1 because there is no comman factor other than 1.
we know that if 1 is the gcf of y and z, so this will mean that gcf is also 1 for x,y,and z.

So sufficient.

D is correct

x,y & z >0 and integers. Calculate HCF:

To calculate HCF we need to know whether x,y or z share any factors amongst them, in case they do not and are thus co-prime then HCF of x,y & z would simply be 1. If we find that 2 of the three integers are co-prime, HCF of all 3 together will also be 1 as they have no factors that are common amongst all 3 integers even if 2 of the integers have common factors.

(1) X = 32, Y = 81 - x & y are co-prime, their HCF should thus be 1 and whether z shares a factor with either x or y or doesn't matter because we know that x and y are co-prime thus \(HCF(x,y,z)=1\). Because a single solution is possible even if value of z is unkown, st-1 alone is sufficient.

(2) Y = 81, Z = 121 - Same as st-1, we know that y & z are co-prime, their HCF should thus be 1 and whether x shares a factor with either y or z or doesn't matter because we know that y and z are co-prime thus \(HCF(x,y,z)=1\). Because a single solution is possible even if value of x is unkown, st-2 alone is sufficient.

Both statements (1) & (2) are individually sufficient and the answer is thus D.

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

(1) X = 32, Y = 81
X = 2x2x2x2x2
Y = 3x3x3x3

X and Y do not have any common factors. So, HCF of X and Y will be 1.

Irrespective of the value of Z, HCF of X, Y, and Z MUST BE 1 because it has to be common for all 3.

(1) IS SUFFICIENT


(2) Y = 81, Z = 121
Y = 3x3x3x3
Z = 11x11

Y and Z do not have any common factors. So, HCF of Y and Z will be 1.

Irrespective of the value of X, HCF of X, Y, and Z MUST BE 1 because it has to be common for all 3.


(2) IS SUFFICIENT

ANSWER: D - Each Is Sufficient

GCD (X,Y,Z)= GCD [GCD(X,Y), Z]= GCD [X, GCD(Y,Z)]

Statement 1-
GCD (X,Y,Z)= GCD [GCD(X,Y), Z]
GCD (32, 81, Z)= GCD [GCD(32, 81), Z]

\(32=2^5\), and \(81= 3^4\)
Hence GCD(32, 81)=1

GCD [GCD(32, 81), Z]= GCD(1, Z)=1 {As, Z is a positive integer.}
Sufficient

Statement 2-
GCD (X,Y,Z)= GCD [X, GCD(Y,Z)]
GCD (X, 81, 121)= GCD [X, GCD(81, 121)]

\(81= 3^4\), and \(121= 11^2\)
Hence GCD(81, 121)=1

GCD [X, GCD(81, 121)]= GCD(X, 1)=1 {As, X is a positive integer.}
Sufficient

IMO D

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).

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

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.

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

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

Solution:

Question Stem Analysis:

We need to find HCF/GCF of 3 positive numbers X, Y & Z
Step 1 : Express each number as a product of prime factors.
Step 2: HCF is the product of all common prime factors using the least power of each common prime factor.
HCF is the product of all common prime factors using the least power of each common prime factor.

Statement One analysis:

X = 32 & Y = 81,
expressing in primes X = \(2^5\) & Y =\(3^4\)
Since there are no common factors in X & Y, Naturally there won't be any common factors in X Y & Z, For HCF, there must be a factor common in ALL 3 numbers
If there are no common factors, the HCF is 1.
Hence statement one alone is sufficient. Eliminate C & E

Statement two Alone:

Y= 81, Z = 121
Y = \(3^4\) , Z = \(11^2\)
Using the same logic as we used in statement one, Here you can see that there are no common prime factors.

Hence, HCF = 1
Statement two is also sufficient

Hence the answer is D

(1) X=32 = 2^5 and Y=81=3^4. Notice 2^5 and 3^4 have no factors in common other than 1. no matter what value we choose for z, the gcf will =1. Observe: if z=2 gcf still =1 and if z=3 gcf still =1 sufficient

(2) y=81=3^4 and z=121=11^2. Same story here. Y and z have no factors in common so the gcf will be 1 no matter what the value of x. Observe: if x=3 gcf still =1 and if x=11 gcf still =1, any chose of x gives gcf=1 Sufficient

OA is D

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