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Magazine Chapter 25: Problem 15
Is \(\sqrt{m+n}\) an integer? (1) \(\sqrt{m}\) is an integer. (2) \(\sqrt{n}\) is an integer.
Short Answer
Expert verified
Both conditions together are not sufficient to ensure \( \sqrt{m+n} \) is an integer.
Step by step solution
01
Understand the Problem
We need to determine if \( \sqrt{m+n} \) is an integer. We have two conditions given: (1) \( \sqrt{m} \) is an integer; (2) \( \sqrt{n} \) is an integer. Each of these individual statements tells us something about the values of \( m \) and \( n \).
02
Rephrase the Given Statements
Given (1): \( \sqrt{m} \) is an integer implies \( m \) is a perfect square, say \( m = a^2 \) where \( a \) is an integer. Given (2): \( \sqrt{n} \) is an integer implies \( n \) is a perfect square, say \( n = b^2 \) where \( b \) is an integer.
03
Combine Both Conditions
If both conditions are met, then \( m + n = a^2 + b^2 \). We need to determine if \( \sqrt{a^2 + b^2} \) is an integer, meaning \( a^2 + b^2 \) must also be a perfect square.
04
Consider Possible Values
Example: If \( a = 3 \) and \( b = 4 \), then \( m = 9 \), \( n = 16 \), and \( m + n = 25 \). Here \( \sqrt{25} = 5 \), which is an integer. However, if \( a = 1 \) and \( b = 2 \), then \( m = 1 \), \( n = 4 \), and \( m + n = 5 \). Here \( \sqrt{5} \) is not an integer.
05
Evaluate the Necessity of Both Conditions
Both conditions by themselves are necessary but not sufficient to conclude that \( \sqrt{m+n} \) is an integer. \( m+n \) must be specifically checked to confirm it's a perfect square.
06
Sufficiency Conclusion
Neither condition alone guarantees that \( \sqrt{m+n} \) is an integer, nor do both combined guarantee it without further context. Therefore, neither statement alone nor together are sufficient to conclude that \( \sqrt{m+n} \) is an integer.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integer Properties
Understanding integer properties is crucial in solving many math problems, including those you might encounter while preparing for the GMAT. An integer is a whole number that can be either positive, negative, or zero. It does not include fractions or decimals. When working with integers, several properties are particularly useful:
- Closure: The sum or product of any two integers will always be an integer. This means if you add or multiply any integers together, the result is also an integer.
- Identity: Adding zero to any integer results in the same integer. Similarly, multiplying any integer by one results in the same integer.
- Inverse: Every integer has a corresponding inverse. Adding an integer and its inverse (negative counterpart) results in zero.
- Commutative and Associative Laws: When adding or multiplying integers, the order in which you perform the operation does not change the result. The grouping of numbers does not affect the sum or product.
Perfect Squares
Perfect squares are the squares of integers. They emerge when multiplying a number by itself: for example, 1, 4, 9, 16, and so on. Knowing about perfect squares is especially vital when dealing with quadratic equations and square roots.
- Definition: If a number is a perfect square, it is the result of squaring an integer. For instance, if \( m = a^2 \), then \( \sqrt{m} \) is an integer, \( a \).
- Common Perfect Squares: Some frequently used perfect squares are 1, 4, 9, 16, 25, 36, etc. These often come up in GMAT problems.
- Square Roots: The square root of a perfect square \( a^2 \) is the integer \( a \). This means \( \sqrt{a^2} = a \).
Data Sufficiency
Data sufficiency questions are a unique type of problem you will encounter on the GMAT. These questions test not only your calculation skills but also your ability to determine when enough information is available to solve a problem. Here are some key points to keep in mind:
- Understanding the Question: First, thoroughly understand what is being asked. For instance, in this exercise, you need to determine if \( \sqrt{m+n} \) is an integer.
- Analyzing Conditions: Look at each piece of information given to you. Each statement provides a certain amount of insight, and you must discern how much each contributes to answering the question.
- Combining Information: Sometimes you will need to combine the results of multiple statements to get a full picture. However, it's crucial to recognize when combining does and doesn't result in a sufficient answer.
- Testing Sufficiency: After using both statements alone and together, check if you can answer definitively. If further context is required, the data might not be sufficient.
