If a and b are positive integers, what is the value of the product ab ?
(1) The least common multiple of a and b is 48.
(2) The greatest common factor of a and b is 4.
The OA is C .
I am confuse here. May any expert give me some help? Thanks.
If a and b are positive integers, what is the value of . . .
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Statement 1:Vincen wrote:If a and b are positive integers, what is the value of the product ab ?
(1) The least common multiple of a and b is 48.
(2) The greatest common factor of a and b is 4.
Case 1: a=1 and b=48, with the result that the LCM of a and b is 48
In this case, ab = 1*48 = 48.
Case 2: a=2 and b=48, with the result that the LCM of a and b is 48
In this case, ab = 2*48 = 96.
Since ab can be different values, INSUFFICIENT.
Statement 2:
Case 1: a=4 and b=4, with the result that the GCF of a and b is 4
In this case, ab = 4*4 = 16.
Case 2: a=4 and b=8, with the result that the GCF of a and b is 4
In this case, ab = 4*8 = 32.
Since ab can be different values, INSUFFICIENT.
Statements combined:
RULE:
(GCF of x and y)(LCM of x and y) = xy.
Thus:
ab = (GCF of a and b)(LCM of a and b) = 4*48 = 192.
SUFFICIENT.
The correct answer is C.
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Target question: What is the value of the product ab?Vincen wrote:If a and b are positive integers, what is the value of the product ab ?
(1) The least common multiple of a and b is 48.
(2) The greatest common factor of a and b is 4.
Statement 1: The least common multiple of a and b is 48
Let's TEST some numbers.
There are several values of a and b that satisfy statement 1. Here are two:
Case a: a = 1 and b = 48. In this case, ab = (1)(48) = 48
Case b: a = 2 and b = 48. In this case, ab = (2)(48) = 96
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: The greatest common factor (aka divisor) of a and b is 4
Let's TEST some numbers (again).
There are several values of a and b that satisfy statement 2. Here are two:
Case a: a = 8 and b = 4. In this case, ab = (8)(4) = 32
Case b: a = 4 and b = 4. In this case, ab = (4)(4) = 16
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined
--------ASIDE----------------------
There's a nice rule that says:
(greatest common divisor of x and y)(least common multiple of x and y) = xy
Example: x = 10 and y = 15
Greatest common divisor of 10 and 15 = 5
Least common multiple of 10 and 15 = 30
Notice that these values satisfy the above rule, since (5)(30) = (10)(15)
--------BACK TO THE QUESTION! ----------------------
When we apply the above rule, we get: (4)(48) = ab
So, ab = 192
Since we can answer the target question with certainty, the combined statements are SUFFICIENT
Answer: C
Cheers,
Brent
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Solution:
We need to determine the value of the product ab, given that a and b are positive integers.
Statement One Alone:
Knowing that the least common multiple (LCM) of a and b is 48 is not sufficient. For example, if a = 1 and b = 48, then ab = 48. However, if a = 2 and b = 48, then ab = 96.
Statement Two Alone:
Knowing that the greatest common factor (GCF) of a and b is 4 is not sufficient. For example, if a = 4 and b = 8, then ab = 32. However, if a = 4 and b = 12, then ab = 48.
Statements One and Two Together:
Knowing both values of the LCM and GCF of a and is sufficient since we have a formula that relates the two:
LCM(a, b) * GCF(a, b) = a * b
Therefore, ab = 48 * 4 = 192.
Answer: C
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