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If X is the GCD of three positive integers x,y,z whose LCM is Y,then which of the following statements must be true.
A)\(X*Y=x*y*z\)
B)Least possible value of X is z
C)x,y,z are all divisible by Y
D)x,y,z are all divisible by X
E)X must be less than z
A)\(X*Y=x*y*z\)
B)Least possible value of X is z
C)x,y,z are all divisible by Y
D)x,y,z are all divisible by X
E)X must be less than z
28 Nov 2016, 12:45
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stonecold
If X is the GCD (greatest common divisor) of three positive integers x,y,z whose LCM is Y,then which of the following statements must be true.
A)\(X*Y=x*y*z\)
B)Least possible value of X is z
C)x,y,z are all divisible by Y
D)x,y, and z are all divisible by X
E)X must be less than z
A)\(X*Y=x*y*z\)
B)Least possible value of X is z
C)x,y,z are all divisible by Y
D)x,y, and z are all divisible by X
E)X must be less than z
IMPORTANT: If d is a divisor of N, we can also say that N is divisible by d
For example, since 6 is a divisor of 24, we can also say that 24 is divisible by 6
If X is the greatest common divisor (GCD) of x, y, and z, then we know that X is a divisor of x, X is a divisor of y, and X is a divisor of z
So, by the above property, we can say that x, y, and z are all divisible by X
Answer:
Cheers,
Brent
28 Nov 2016, 12:54
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stonecold
If X is the GCD of three positive integers x,y,z whose LCM is Y,then which of the following statements must be true.
A) XY = xyz
B) Least possible value of X is z
C) x,y,z are all divisible by Y
D) x,y,z are all divisible by X
E) X must be less than z
A) XY = xyz
B) Least possible value of X is z
C) x,y,z are all divisible by Y
D) x,y,z are all divisible by X
E) X must be less than z
In my above post, I showed why D is the correct answer.
We can also try showing why the other answer choices are incorrect.
A) XY = xyz
If x = 2, y = 2 and z = 2, then X = 2 and Y = 2. In this case XY = (2)(2) = 4, and xyz = (2)(2)(2) = 8
So, we can see that XY ≠ xyz
So, this statement need not be true.
B) Least possible value of X is z
If x = 3, y = 2 and z = 2, then X = 1 and Y = 6.
In this case, X is less than z
So, this statement need not be true.
C) x,y,z are all divisible by Y
If x = 3, y = 2 and z = 2, then X = 1 and Y = 6.
In this case, x, y and z are NOT divisible by Y (e.g., 3 is not divisible by 6)
So, this statement need not be true.
E) X must be less than z
If x = 2, y = 2 and z = 2, then X = 2 and Y = 2. In this case XY = (2)(2) = 4, and xyz = (2)(2)(2) = 8
So, we can see that X is NOT less than z
So, this statement need not be true.
By the process of elimination, we can see that the correct answer is D.
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