Two Integers, N And P, Have A Product Of 24. What Is The Lowest Possible Sum Of N And P? Explain/show
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Understanding the relationship between multiplication and addition is fundamental in mathematics, especially when working with integers. The problem of finding two integers, N and P, such that their product is a specific number—in this case, 24—and determining the minimum possible sum of these integers is a classic example of solving equations with given constraints.
This article delves into this problem, exploring the various integer pairs that satisfy the product condition and identifying which pair yields the lowest sum. Along the way, we will understand the importance of factors, divisors, and the role of positive and negative numbers in such problems. Whether you're a student seeking to improve your algebra skills or a math enthusiast, this comprehensive guide will clarify the problem-solving process and provide clear explanations to enhance your understanding.
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Understanding the Problem
The problem states:
> Two integers, N and P, have a product of 24. What is the lowest possible sum of N and P? Explain/show.
Let's break down what this means:
- Two integers, N and P: These are the two numbers we're working with.
- Product of 24: N × P = 24.
- Lowest possible sum: Among all pairs (N, P) that satisfy the product condition, find the pair where N + P is minimized.
This problem involves two main tasks:
- Identifying all pairs of integers whose product is 24.
- Calculating their sums and determining which sum is the smallest.
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Factors of 24 and Possible Integer Pairs
To find all pairs of integers whose product is 24, we need to understand the concept of factors.
What Are Factors?
Factors are numbers that evenly divide another number. For example, the factors of 24 are numbers that divide 24 without leaving a remainder. These are:
- 1, 2, 3, 4, 6, 8, 12, 24
Since the product of the two integers is 24, the pairs (N, P) are essentially the factor pairs of 24.
Positive Factor Pairs of 24
The positive factor pairs of 24 are:
- (1, 24)
- (2, 12)
- (3, 8)
- (4, 6)
Because multiplication is commutative, these pairs are interchangeable; for example, (1, 24) and (24, 1) are the same in terms of their product.
Negative Factor Pairs of 24
Since negative numbers multiply to a positive number when both are negative, the negative factor pairs of 24 are:
- (-1, -24)
- (-2, -12)
- (-3, -8)
- (-4, -6)
These pairs also satisfy the product condition because multiplying two negatives yields a positive.
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Calculating the Sum of Each Pair
Now, we'll compute the sum for each pair and identify which sum is the lowest.
Positive pairs and their sums
| Pair | Sum (N + P) |
|---------|--------------|
| (1, 24) | 1 + 24 = 25 |
| (2, 12) | 2 + 12 = 14 |
| (3, 8) | 3 + 8 = 11 |
| (4, 6) | 4 + 6 = 10 |
Negative pairs and their sums
| Pair | Sum (N + P) |
|---------|--------------|
| (-1, -24) | -1 + (-24) = -25 |
| (-2, -12) | -2 + (-12) = -14 |
| (-3, -8) | -3 + (-8) = -11 |
| (-4, -6) | -4 + (-6) = -10 |
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Identifying the Lowest Sum
In the context of the problem, the lowest sum refers to the smallest numerical value, which could be negative or positive depending on the pairs.
- Among the positive pairs, the smallest sum is 10 (from the pair (4, 6)).
- Among the negative pairs, the smallest sum is -25 (from the pair (-1, -24)).
Since the question asks for the lowest possible sum, and considering that negative sums are less than positive sums, the pair (-1, -24) yields the lowest sum of -25.
Therefore, the pair (-1, -24) satisfies the condition of having a product of 24 and yields the lowest possible sum, which is -25.
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Conclusion: Final Answer and Explanation
The lowest possible sum of two integers N and P with a product of 24 is -25, achieved when N = -1 and P = -24.
This conclusion is based on examining all factor pairs of 24, including negative pairs, and calculating their sums. Negative pairs produce sums that are less than any positive pair because adding two negative numbers results in a more negative sum.
This problem illustrates the importance of considering both positive and negative factors when dealing with integer equations. It also demonstrates how understanding factors and their properties can lead to efficient solutions in algebraic problems.
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Additional Insights and Tips for Similar Problems
- Always consider both positive and negative factors when solving for integer solutions involving multiplication.
- To find the pair with the lowest sum, look for negative factor pairs first because they tend to produce the smallest (most negative) sums.
- Remember that the absolute value of the factors influences the sum; larger absolute values tend to produce more negative sums when both factors are negative.
- Practice with different products to strengthen your understanding of factor pairs and their sums.
Summary
| Step | Explanation |
|---------|--------------|
| 1. Find all factor pairs of 24 (positive and negative). | (1, 24), (2, 12), (3, 8), (4, 6) and their negatives. |
| 2. Calculate the sum for each pair. | Compute N + P for each pair. |
| 3. Identify the smallest sum. | Negative pairs produce sums less than positive pairs; (-1, -24) yields -25. |
| 4. Conclusion. | The pair (-1, -24) gives the lowest sum: -25. |
This problem exemplifies key algebraic concepts such as factors, divisibility, and the importance of considering negative solutions to find optimal answers in mathematical problems involving integers.
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By understanding these principles and steps, students and math enthusiasts can confidently approach similar problems involving products and sums of integers.
