Consider all positive integers a, b, and c such that 135 is a factor of each of the expressions 15a,...

1. Translate the problem requirements: We need to find all positive integers a, b, c where 135 divides each of \(15\mathrm{a}\), \(25\mathrm{b}\), and \(30\mathrm{c}\), then determine the GCD of all possible values of \(\mathrm{a} + \mathrm{b} + \mathrm{c}\)
2. Analyze the divisibility conditions: Break down what it means for 135 to be a factor of each expression by finding the prime factorization and determining the minimum requirements for a, b, and c
3. Find the minimum values: Calculate the smallest possible values of a, b, and c that satisfy the divisibility conditions
4. Identify the pattern for all valid values: Determine how a, b, and c can vary beyond their minimum values while maintaining the divisibility requirement
5. Calculate the GCD of all possible sums: Use the pattern to find what factor is common to every possible value of \(\mathrm{a} + \mathrm{b} + \mathrm{c}\)

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what this problem is asking in plain English. We have three expressions: \(15\mathrm{a}\), \(25\mathrm{b}\), and \(30\mathrm{c}\), where a, b, and c are positive integers. The problem tells us that 135 must be a factor of each of these expressions.

What does it mean for 135 to be a factor? It means that when we divide \(15\mathrm{a}\) by 135, we get a whole number. Same for \(25\mathrm{b} \div 135\) and \(30\mathrm{c} \div 135\).

Our goal is to find all possible values of a, b, and c that make this work, then look at all the possible sums \(\mathrm{a} + \mathrm{b} + \mathrm{c}\), and finally find what number divides ALL of these sums.

Process Skill: TRANSLATE - Converting the factor relationship into clear mathematical requirements

2. Analyze the divisibility conditions

To understand when 135 divides our expressions, let's first break down 135 into its basic building blocks (prime factors):
\(135 = 27 \times 5 = 3^3 \times 5^1\)

Now let's look at each expression:

• For \(15\mathrm{a}\): We need 135 to divide \(15\mathrm{a}\). Since \(15 = 3^1 \times 5^1\), we need \(15\mathrm{a}\) to contain at least \(3^3 \times 5^1\)
• For \(25\mathrm{b}\): We need 135 to divide \(25\mathrm{b}\). Since \(25 = 5^2\), we need \(25\mathrm{b}\) to contain at least \(3^3 \times 5^1\)
• For \(30\mathrm{c}\): We need 135 to divide \(30\mathrm{c}\). Since \(30 = 2 \times 3^1 \times 5^1\), we need \(30\mathrm{c}\) to contain at least \(3^3 \times 5^1\)

This tells us what a, b, and c must contain as factors.

Process Skill: INFER - Drawing conclusions about the required prime factor composition

3. Find the minimum values

Let's figure out the smallest possible values for a, b, and c:

For \(15\mathrm{a}\) to be divisible by 135:

• \(15\mathrm{a}\) must contain \(3^3 \times 5^1\)
• Since \(15 = 3^1 \times 5^1\), we need a to provide the missing factors: \(3^2 = 9\)
• So the minimum value of a is 9

For \(25\mathrm{b}\) to be divisible by 135:

• \(25\mathrm{b}\) must contain \(3^3 \times 5^1\)
• Since \(25 = 5^2\), we need b to provide \(3^3 = 27\)
• So the minimum value of b is 27

For \(30\mathrm{c}\) to be divisible by 135:

• \(30\mathrm{c}\) must contain \(3^3 \times 5^1\)
• Since \(30 = 2 \times 3^1 \times 5^1\), we need c to provide \(3^2 = 9\)
• So the minimum value of c is 9

Let's verify: \(15(9) = 135\) ✓, \(25(27) = 675 = 135 \times 5\) ✓, \(30(9) = 270 = 135 \times 2\) ✓

4. Identify the pattern for all valid values

Now that we know the minimum values, what about all other possible values?

Since we need:

• a to be at least 9, all valid values of a are: 9, 18, 27, 36, ... = \(9\mathrm{k}\) where k is any positive integer
• b to be at least 27, all valid values of b are: 27, 54, 81, 108, ... = \(27\mathrm{m}\) where m is any positive integer
• c to be at least 9, all valid values of c are: 9, 18, 27, 36, ... = \(9\mathrm{n}\) where n is any positive integer

Therefore, any valid sum \(\mathrm{a} + \mathrm{b} + \mathrm{c}\) has the form:
\(\mathrm{a} + \mathrm{b} + \mathrm{c} = 9\mathrm{k} + 27\mathrm{m} + 9\mathrm{n} = 9(\mathrm{k} + 3\mathrm{m} + \mathrm{n})\)

Since k, m, and n can be any positive integers, \((\mathrm{k} + 3\mathrm{m} + \mathrm{n})\) can be any integer ≥ 5 (when k=1, m=1, n=1).

5. Calculate the GCD of all possible sums

Every possible sum \(\mathrm{a} + \mathrm{b} + \mathrm{c}\) can be written as 9 times some integer that's at least 5.

The possible sums are: \(9 \times 5, 9 \times 6, 9 \times 7, 9 \times 8, 9 \times 9, 9 \times 10, ...\)
Which gives us: 45, 54, 63, 72, 81, 90, ...

What's the greatest number that divides ALL of these values? Since each one is 9 times an integer, and we can get consecutive integers starting from 5, the only common factor is 9.

To confirm: \(\mathrm{GCD}(45, 54, 63, 72, 81, ...) = 9\)

6. Final Answer

The greatest common factor of all possible sums \(\mathrm{a} + \mathrm{b} + \mathrm{c}\) is 9.

The answer is C.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "135 is a factor of" as "135 equals"

Students may confuse the phrase "135 is a factor of \(15\mathrm{a}\)" and think it means \(15\mathrm{a} = 135\), leading them to solve \(\mathrm{a} = 9\) as the only solution rather than understanding that \(15\mathrm{a}\) must be a multiple of 135. This fundamental misunderstanding would completely derail their approach from the start.

2. Focusing on finding specific values instead of the general pattern

Students might get stuck trying to find all individual possible values of a, b, and c rather than recognizing they need to find the pattern that describes ALL possible values. They may list out a few cases like (9,27,9), (18,27,9), etc., but miss the systematic approach of expressing these as multiples.

3. Misunderstanding what "greatest common factor of all possible sums" means

Students may interpret this as finding the GCD of just a few example sums rather than understanding they need to find what divides EVERY possible sum \(\mathrm{a} + \mathrm{b} + \mathrm{c}\). This leads them to calculate GCD of specific cases like \(\mathrm{GCD}(45, 54, 63)\) instead of recognizing the infinite set of all possible sums.

Errors while executing the approach

1. Incorrect prime factorization or factor analysis

Students may make errors when breaking down \(135 = 3^3 \times 5^1\) or when analyzing what additional factors a, b, and c need to provide. For example, they might incorrectly conclude that a needs \(3^3\) instead of \(3^2\) as a factor, leading to wrong minimum values.

2. Arithmetic errors in finding minimum values

When calculating the minimum values, students might make computational mistakes. For instance, they might calculate that b needs to provide \(3^3 = 27\) but then incorrectly conclude the minimum value is something other than 27, or make errors in verification steps.

3. Incorrect algebraic manipulation when expressing the general form

Students may struggle with correctly expressing \(\mathrm{a} + \mathrm{b} + \mathrm{c} = 9\mathrm{k} + 27\mathrm{m} + 9\mathrm{n} = 9(\mathrm{k} + 3\mathrm{m} + \mathrm{n})\). They might factor incorrectly or fail to recognize that this means every sum is a multiple of 9.

Errors while selecting the answer

1. Selecting the minimum sum instead of the GCD

Students might calculate the minimum possible sum (which is 45) and mistakenly select an answer choice that represents this value or a factor of it, rather than recognizing that 9 is the greatest common divisor of ALL possible sums.

2. Choosing a factor of the correct answer

After correctly determining that all sums are multiples of 9, students might select 3 (choice A) thinking that since 3 divides 9, it's also a common factor, without recognizing that 9 is the GREATEST common factor.