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Find all possible values of b, where (a,b,c) satisfies \[abc=3abc=4abc=5\] and a, b, and c are positive. Enter all values, separated by commas. 

 Oct 12, 2024
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We are tasked with finding all possible values of b that satisfy the following system of equations involving the floor functions:

 

abc=3


abc=4


abc=5

 

where a, b, and c are positive real numbers. We will solve this step by step by exploring possible values for a, b, and c.

 

### Solution By Steps

 

#### Step 1: Analyze the first equation


From Equation (1):

 

abc=3

 

The floor function a represents the greatest integer less than or equal to a. Since a is positive, the possible values of a are integers.

 

We first assume a=1, which is the smallest possible value because a>0. Substituting this into Equation (1):

 

1bc=3

 

Thus, we have:

 

bc=3

 

#### Step 2: Analyze the second equation


Now, consider Equation (2):

 

abc=4

 

We already know bc=3 from Equation (4). To satisfy this equation, we explore possible values of b.

 

Let’s assume b=1 first. Substituting into Equation (2):

 

a1c=4

 

So,

 

ac=4

 

#### Step 3: Analyze the third equation


Now consider Equation (3):

 

abc=5

 

We already have bc=3 and ac=4. To satisfy this equation, we explore possible values of c.

 

Let’s assume c=1. Substituting into Equation (3):

ab1=5

 

So:

 

ab=5

 

#### Step 4: Solve the system of equations


Now, we have three equations:

 

1. bc=3


2. ac=4


3. ab=5

 

We can solve this system step by step. First, solve for c from Equation (5):

 

c=4a

 

Substitute this into Equation (4):

 

b4a=3

 

Simplifying:

 

b=3a4

 

Now substitute this into Equation (6):

 

a3a4=5

 

Simplifying:

 

3a24=5

 

Multiply both sides by 4:

 

3a2=20

 

Solve for a2:

 

a2=203

 

So:

 

a=203=2153

 

#### Step 5: Find the value of b


Now that we have a=2153, substitute this back into the expression for b:

 

b=3a4=3×21534=2154=152

 

Thus, the value of b is:

b=152

 Oct 12, 2024

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