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If \(\mathrm{m}\) is a positive integer and \(\mathrm{f}\) and \(\mathrm{g}\) are factors of \(\mathrm{m}\), which of the following must be an integer?
Let's start by understanding what we're being asked in everyday language.
We have a positive integer m, and f and g are factors of m. This means that when we divide m by f, we get a whole number with no remainder. Same thing with g - when we divide m by g, we get a whole number.
The question asks: which of these three expressions must always be an integer, no matter what factors f and g we choose?
The key word here is "must" - we need to find expressions that are always integers, regardless of which factors we pick.
Process Skill: TRANSLATE - Converting the mathematical language into plain understanding
Let's use a simple example to test each expression. Let's say \(\mathrm{m = 12}\).
The factors of 12 are: 1, 2, 3, 4, 6, and 12. Let's pick \(\mathrm{f = 3}\) and \(\mathrm{g = 4}\) as our factors.
With \(\mathrm{m = 12}\), \(\mathrm{f = 3}\), \(\mathrm{g = 4}\):
\(\mathrm{m/(f+g) = 12/(3+4) = 12/7 = 1.714...}\)
This is not an integer! So Expression I doesn't always have to be an integer.
With the same values:
\(\mathrm{m/(fg) = 12/(3 \times 4) = 12/12 = 1}\)
This gave us an integer, but let's try different factors to see if it always works.
Let's try \(\mathrm{f = 2}\) and \(\mathrm{g = 3}\) (both factors of 12):
\(\mathrm{m/(fg) = 12/(2 \times 3) = 12/6 = 2}\)
Still an integer. But wait, let's try \(\mathrm{f = 1}\) and \(\mathrm{g = 2}\):
\(\mathrm{m/(fg) = 12/(1 \times 2) = 12/2 = 6}\)
Hmm, these are all working. But let me try a case that might not work. Let's try \(\mathrm{f = 6}\) and \(\mathrm{g = 4}\):
\(\mathrm{m/(fg) = 12/(6 \times 4) = 12/24 = 1/2 = 0.5}\)
Aha! This is not an integer. So Expression II doesn't always have to be an integer either.
With \(\mathrm{f = 3}\), \(\mathrm{g = 4}\):
\(\mathrm{f/g = 3/4 = 0.75}\)
This is clearly not an integer. So Expression III doesn't always have to be an integer.
Let's verify our counterexamples are valid and look for any patterns we might have missed.
Our counterexample: \(\mathrm{m = 12}\), \(\mathrm{f = 3}\), \(\mathrm{g = 4}\)
Check: Is 3 a factor of 12? Yes, \(\mathrm{12 ÷ 3 = 4}\)
Check: Is 4 a factor of 12? Yes, \(\mathrm{12 ÷ 4 = 3}\)
Result: \(\mathrm{12/(3+4) = 12/7 ≈ 1.71}\) (not an integer)
This counterexample is valid.
Our counterexample: \(\mathrm{m = 12}\), \(\mathrm{f = 6}\), \(\mathrm{g = 4}\)
Check: Is 6 a factor of 12? Yes, \(\mathrm{12 ÷ 6 = 2}\)
Check: Is 4 a factor of 12? Yes, \(\mathrm{12 ÷ 4 = 3}\)
Result: \(\mathrm{12/(6 \times 4) = 12/24 = 1/2 = 0.5}\) (not an integer)
This counterexample is valid.
Our counterexample: \(\mathrm{f = 3}\), \(\mathrm{g = 4}\) (both factors of 12)
Result: \(\mathrm{3/4 = 0.75}\) (not an integer)
This counterexample is valid.
Process Skill: CONSIDER ALL CASES - Systematically testing different factor combinations to ensure our conclusions are sound
Based on our systematic testing, we found counterexamples for all three expressions:
Since we found valid counterexamples showing that none of the three expressions must always be integers, the answer is that none of them must be an integer.
The correct answer is A. None
We demonstrated through concrete examples that each of the three expressions can result in non-integer values even when f and g are valid factors of m. Therefore, none of the expressions must always be an integer.
Faltering Point 1: Misinterpreting "must be an integer"
Students often confuse "must be an integer" with "can be an integer." They may think that if they find one example where an expression equals an integer, then it "must" always be an integer. The word "must" means the expression has to be an integer for ALL possible factor pairs, not just some cases. This leads students to incorrectly conclude that expressions are always integers based on limited testing.
Faltering Point 2: Not understanding what "factors" means
Students may think that f and g have to be different numbers, or that they must be prime factors, or that fg must equal m. The question simply states that f and g are factors of m, which means f divides m evenly AND g divides m evenly. Students need to realize that any factor of m can be chosen for f and g, including the same factor being used twice (like \(\mathrm{f = g = 3}\)), and factors like 1 and m itself.
Faltering Point 3: Choosing overly simple examples initially
Students often start with very basic examples like \(\mathrm{m = 6}\) with \(\mathrm{f = 2}\), \(\mathrm{g = 3}\), or use factors that have special relationships. This can make expressions seem to work when they don't always work. A better approach is to deliberately try examples that might break the pattern, such as using larger factors, factors where \(\mathrm{fg > m}\), or factors that don't multiply to give m.
Faltering Point 1: Stopping after finding the first integer result
Students may test Expression II with factors like \(\mathrm{f = 3}\), \(\mathrm{g = 4}\) for \(\mathrm{m = 12}\), get \(\mathrm{m/(fg) = 12/12 = 1}\), and conclude it always works. They don't push further to test cases where fg might be larger than m (like \(\mathrm{f = 6}\), \(\mathrm{g = 4}\) for \(\mathrm{m = 12}\)), which would give non-integer results. The key is testing multiple factor combinations, especially edge cases.
Faltering Point 2: Arithmetic errors in fraction calculations
When calculating expressions like \(\mathrm{12/7}\) or \(\mathrm{12/24}\), students may make computational mistakes or incorrectly convert to decimals. For example, they might think \(\mathrm{12/24 = 2}\) instead of \(\mathrm{1/2}\), or miscalculate \(\mathrm{12/7}\). These errors can lead to wrong conclusions about whether expressions yield integers.
Faltering Point 3: Invalid factor selection
Students may accidentally choose numbers that aren't actually factors of m. For example, if \(\mathrm{m = 12}\), they might test with \(\mathrm{f = 5}\), not realizing that 5 doesn't divide 12 evenly. This leads to testing invalid scenarios and potentially wrong conclusions about the expressions.
Faltering Point 1: Confusing "None" with "All"
After finding that each expression can sometimes be non-integer, students might mistakenly think this means "all expressions fail" and look for an option like "I, II, and III are never integers" rather than understanding that "None" means "none of them MUST be integers." The correct interpretation is that since each expression can fail to be an integer, none of them are required to always be integers.
Faltering Point 2: Partial credit thinking
Students may find that Expression II works in most cases they test, and think it deserves partial credit even though they found one counterexample. They might select "II only" thinking that since it works "most of the time," it should count. However, the question asks what MUST be integers, meaning they work ALL the time - even one counterexample disqualifies an expression completely.
Step 1: Choose a concrete value for m
Let's use \(\mathrm{m = 12}\), which has multiple factor pairs that will help us test all three expressions systematically.
Step 2: Identify factor pairs of 12
The factors of 12 are: 1, 2, 3, 4, 6, 12
We can use different factor pairs (f, g) to test each expression.
Step 3: Test Expression I: \(\mathrm{m/(f+g)}\)
Using \(\mathrm{f = 2}\), \(\mathrm{g = 3}\) (both factors of 12):
\(\mathrm{m/(f+g) = 12/(2+3) = 12/5 = 2.4}\)
Since 2.4 is not an integer, Expression I does not always yield an integer.
Step 4: Test Expression II: \(\mathrm{m/(fg)}\)
Using \(\mathrm{f = 2}\), \(\mathrm{g = 3}\):
\(\mathrm{m/(fg) = 12/(2 \times 3) = 12/6 = 2}\)
This gives an integer, but let's try another pair.
Using \(\mathrm{f = 2}\), \(\mathrm{g = 6}\):
\(\mathrm{m/(fg) = 12/(2 \times 6) = 12/12 = 1}\)
Using \(\mathrm{f = 3}\), \(\mathrm{g = 6}\):
\(\mathrm{m/(fg) = 12/(3 \times 6) = 12/18 = 2/3}\)
Since \(\mathrm{2/3}\) is not an integer, Expression II does not always yield an integer.
Step 5: Test Expression III: \(\mathrm{f/g}\)
Using \(\mathrm{f = 2}\), \(\mathrm{g = 3}\):
\(\mathrm{f/g = 2/3}\)
Since \(\mathrm{2/3}\) is not an integer, Expression III does not always yield an integer.
Step 6: Conclusion
All three expressions can produce non-integer results when we use valid factors of m. Therefore, none of the expressions must be an integer.
Answer: A. None