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Please help, given a, b, and c are positive integers greater than 2 what are all the possible values of a*b*c given a*b*c=5(a-2)(b-2)(c-2).

 Dec 17, 2023
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Here's how we can approach this:

 

1. Factorization:

First, we can factor the expression on the right side of the equation:

 

5(a-2)(b-2)(c-2) = 5 * 2 * 2 * (a-2) * (b-2) * (c-2) = 20 * (a-2) * (b-2) * (c-2)

 

2. Observation:

 

Now, we observe that the product abc appears on the left side of the equation, while each factor (a-2), (b-2), and (c-2) appears on the right side multiplied by 20. This suggests that each of these factors must be a divisor of abc.

 

3. Finding divisors:

 

Since a, b, and c are positive integers greater than 2, their possible divisors are:

 

For a: 3, 4, 5, 6, 10, 12, ...

For b: 3, 4, 5, 6, 10, 12, ...

For c: 3, 4, 5, 6, 10, 12, ...

 

4. Identifying valid combinations:

 

We need to find combinations of divisors from each list that, when multiplied together, give a product equal to 20. These combinations will correspond to the possible values of abc.

 

For example, (3, 4, 5) is a valid combination because 3 * 4 * 5 = 60, and 60 / 20 = 3 (which satisfies the equation). Similarly, (5, 2, 2) is also valid because 5 * 2 * 2 = 20.

 

5. Listing all possibilities:

 

By systematically checking all combinations of divisors for a, b, and c, we can find all possible values of abc:

 

3 * 4 * 5 = 60

5 * 2 * 2 = 20

4 * 3 * 3 = 36

6 * 2 * 2 = 24

10 * 1 * 2 = 20

12 * 1 * 1 = 12

 

Therefore, all possible values of abc are: 60, 20, 36, 24, and 12.

 Dec 17, 2023

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