If a and b are positive integers, a b, the greatest common factor of

Question

If a and b are positive integers, a > b, the greatest common factor of a and b is x, and the least common multiple of a and b is y, then what is the least common multiple of (a - b) and b ?

A. x
B. y
C. (a - b)b
D. (a - b)b/x
E. x(a - b)

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TestPrepUnlimited · Answer

First, we have the formula  LCM = \frac{a*b }{ GCF} .

x is a factor of both a and b, so (a - b) still has a factor of x. Then the GCF of (a - b) and (b) is still x. Using the formula above we have  LCM = \frac{(a-b)*b }{ x}

Ans: D

ankita1905 · Answer

RULE : If HCF(a, b) = x => HCF(a, a-b) = HCF(b, a-b) = x

For example, if a=12 and b=18 , then HCF(12,18) = 6 
Also, HDF(12, 18-12) = HCF(18, 18-12) = 6

Can also be understood in the following manner :
HCF(a, b) = x
Therefore, a = x*p and b = x*q where p!=q and HCF(p, q) = 1
Further, HCF(a, a - b) = HCF(x*p, x *(p -q)) = x

Objective : LCM(a-b, b) = ?

FORMULA : a * b = HCF(a, b) * LCM(a, b) 
From above, HCF(b, a-b) = x,
Therefore, LCM(a-b, b) =  /x

Answer : Option D

fireagablast · Answer

a = 3*5
b = 2*5
x=5

a-b = 5

GCF(a-b, b) = GCF(5, 10) = 10

option D --> (a-b)*b/x = 5*10/5 = 10

gurmukh · Answer

Let a =5×5
 b =4×5
Therefore a-b =5
 x = 5
 y =4×5×5=100
LCM of (a-b,b)
LCM of (5, 20) =20
 (a-b)b/x = 5×20/5=20
Option D is the answer

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gurmukh · Answer

The problem with 20 and 10 is how will you differentiate between option A and D both of them are comming correct.

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ScottTargetTestPrep · Answer

We can let a = 5 and b = 2. So x = GCF(5, 2) = 1, and y = LCM(5, 2) = 10. Since a - b = 5 - 2 = 3, we have LCM(3, 2) = 6. We can check which answer choice will also give us the value of 6.

A. x = 1 → This is not 6.

B. y = 10 → This is not 6.

C. (a - b)b = (5 - 2)(2) = 6 → This is 6, but we need to make sure the last two choices aren’t 6, either.

D. (a - b)b/x = (5 - 2)(2)/1 = 6 → This is also 6.

E. x(a - b) = 1(5 - 2) = 3 → This is not 6.

We see that both C and D give us 6. To determine which one is the correct answer, we need to use a different pair of values of a and b with a GCF greater than 1. Letting a = 6 and b = 4, we see that x = GCF(6, 4) = 2 and y = LCM(6, 4) = 12. Since a - b = 6 - 4 = 2, we have LCM(2, 4) = 4. We can check which answer choice (between C and D) will also give us the value of 4.

C. (a - b)b = (6 - 4)(4) = 8 → This is not 4.

D. (a - b)b / x = (6 - 4)(4)/2 = 8/2 = 4 → This is 4.

Answer: D

lacktutor · Answer

Let a and b be equal to 15 and 10 respectively.
(a < b)

GCF(a, b)= GCF(15, 10) = x=5
LCM(a, b)= LCM(15, 10) = y=30
---> LCM (a-b, b)= LCM (15-10, 10) = 10

Only answer option D  \frac{(a-b)b}{x} = \frac{(15-10)*10}{5}  gives the same answer.

Answer (D)

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If a and b are positive integers, a > b, the greatest common factor of

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