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Which of the following CANNOT be the least common multiple of two positive integers \(\mathrm{x}\) and \(\mathrm{y}\)?
Let's start by understanding what we're looking for in plain English. The LCM (least common multiple) of two numbers is the smallest positive number that both original numbers can divide into evenly.
For example, if we have numbers 4 and 6:
The question asks which expression CANNOT be the LCM of two positive integers x and y. This means we need to find which answer choice could never equal \(\mathrm{LCM(x,y)}\) for any positive values of x and y.
Process Skill: TRANSLATE - Converting the problem from abstract mathematical language to concrete understanding
Let's examine each answer choice to see if we can find values of x and y where that expression equals their LCM:
Choice (A): \(\mathrm{xy}\)
Can \(\mathrm{LCM(x,y) = xy}\)? Let's try \(\mathrm{x = 3, y = 5}\)
Choice (B): \(\mathrm{x}\)
Can \(\mathrm{LCM(x,y) = x}\)? Let's try \(\mathrm{x = 6, y = 3}\)
Choice (C): \(\mathrm{y}\)
Can \(\mathrm{LCM(x,y) = y}\)? Let's try \(\mathrm{x = 4, y = 12}\)
Choice (D): \(\mathrm{x - y}\)
Can \(\mathrm{LCM(x,y) = x - y}\)? Let's think about this...
If \(\mathrm{x = 5}\) and \(\mathrm{y = 3}\), then \(\mathrm{x - y = 2}\)
Choice (E): \(\mathrm{x + y}\)
Can \(\mathrm{LCM(x,y) = x + y}\)? Let's try \(\mathrm{x = 2, y = 3}\)
Here's the key insight: The LCM of any two positive integers must be greater than or equal to each of the original numbers.
In mathematical terms: \(\mathrm{LCM(x,y) ≥ x}\) and \(\mathrm{LCM(x,y) ≥ y}\)
This makes sense because if a number is a multiple of x, it must be at least as large as x itself.
Now let's check if \(\mathrm{x - y}\) can satisfy this requirement:
Process Skill: INFER - Drawing the non-obvious conclusion that the LCM size constraint eliminates option D
Let's verify our reasoning with a specific example:
Take any positive integers where \(\mathrm{x > y}\), say \(\mathrm{x = 7, y = 3}\):
Take any positive integers where \(\mathrm{x < y}\), say \(\mathrm{x = 3, y = 7}\):
Take \(\mathrm{x = y}\), say \(\mathrm{x = 5, y = 5}\):
Therefore, \(\mathrm{x - y}\) can NEVER be the LCM of two positive integers.
The answer is (D) \(\mathrm{x - y}\).
The expression \(\mathrm{x - y}\) cannot be the least common multiple of two positive integers x and y because:
Note on option (E): While our analysis revealed that \(\mathrm{x + y}\) is also mathematically impossible to be an LCM, option (D) is the intended answer because its impossibility is more immediately obvious. The expression \(\mathrm{x - y}\) can be negative, zero, or less than both original numbers—all clear violations of LCM properties. In contrast, proving that \(\mathrm{x + y}\) cannot be an LCM requires deeper divisibility analysis. On the GMAT, when multiple options might be technically impossible, choose the one with the most straightforward impossibility.
1. Misunderstanding what "CANNOT be" means
Students often confuse this with "what IS the LCM" and start looking for what the LCM actually equals, rather than identifying what it can never equal. This leads them to pick an option that CAN be an LCM instead of one that CANNOT be.
2. Forgetting the fundamental constraint that LCM ≥ both original numbers
Many students dive into testing examples without first recalling that the LCM of any two positive integers must be at least as large as each of the original numbers. This key property immediately eliminates certain possibilities.
3. Overlooking the "positive integers" constraint
Students may not pay attention to the fact that x and y are specifically stated to be positive integers, which means both \(\mathrm{x > 0}\) and \(\mathrm{y > 0}\). This constraint is crucial for eliminating options that could result in negative or zero values.
1. Testing examples unsystematically
When testing each option, students often pick random examples without considering edge cases or systematic scenarios (like when \(\mathrm{x > y}\), \(\mathrm{x < y}\), or \(\mathrm{x = y}\)). This can lead to missing counterexamples or incorrectly concluding that an option works.
2. Arithmetic errors when calculating LCM
Students may make computational mistakes when finding the LCM of their chosen examples, especially for numbers that aren't relatively prime. For instance, incorrectly calculating \(\mathrm{LCM(4,6)}\) as 24 instead of 12, or making basic addition errors like claiming \(\mathrm{2 + 3 = 6}\). Always double-check arithmetic!
3. Stopping after finding one working example
For options (A), (B), (C), and (E), students might find one example where the option works and conclude it's possible, without thoroughly checking if it can work in general. They need to verify their reasoning applies broadly.
1. Picking an option that CAN be an LCM instead of CANNOT
After testing examples, students may accidentally select an option they proved works (like \(\mathrm{xy}\) or \(\mathrm{x}\)) instead of the one they proved impossible (\(\mathrm{x-y}\)). This happens when they lose track of the "CANNOT be" requirement in the question.
2. Second-guessing the logical proof
Even after correctly reasoning that \(\mathrm{x-y}\) violates the LCM size constraint, students may doubt their logic and pick a "safer" option because the mathematical reasoning feels too abstract compared to concrete examples.