Answer :
Certainly! Let's solve the problem step-by-step.
### Part 1: The product of two integers is -20
We need to find two integers whose product is [tex]\(-20\)[/tex] and determine the largest possible value of their sum.
To begin, list the factor pairs of [tex]\(-20\)[/tex]:
1. [tex]\((1, -20)\)[/tex]
2. [tex]\((-1, 20)\)[/tex]
3. [tex]\((2, -10)\)[/tex]
4. [tex]\((-2, 10)\)[/tex]
5. [tex]\((4, -5)\)[/tex]
6. [tex]\((-4, 5)\)[/tex]
Now, calculate the sum for each pair:
1. [tex]\(1 + (-20) = -19\)[/tex]
2. [tex]\(-1 + 20 = 19\)[/tex]
3. [tex]\(2 + (-10) = -8\)[/tex]
4. [tex]\(-2 + 10 = 8\)[/tex]
5. [tex]\(4 + (-5) = -1\)[/tex]
6. [tex]\(-4 + 5 = 1\)[/tex]
Among these, the largest sum is [tex]\(19\)[/tex], which occurs for the pair [tex]\((-1, 20)\)[/tex].
### Part 2: The product of two integers is [tex]\(-3a\)[/tex]
We need to find two integers whose product is [tex]\(-3a\)[/tex] and determine the largest possible value of their sum. Here, [tex]\(a\)[/tex] is a variable representing any positive integer.
For this more general case, let’s consider various factor pairs of [tex]\(-3a\)[/tex]:
Since [tex]\(-3a\)[/tex] has factor pairs [tex]\((p, \frac{-3a}{p})\)[/tex], the sum of these pairs would be:
[tex]\[ p + \frac{-3a}{p} \][/tex]
To maximize this sum, consider maximizing [tex]\(p\)[/tex] and minimizing [tex]\(\frac{-3a}{p}\)[/tex] in a balanced way.
#### Examples using specific values of [tex]\(a\)[/tex]:
- When [tex]\(a=1\)[/tex], [tex]\(-3a = -3\)[/tex]:
1. [tex]\((1, -3)\)[/tex]: [tex]\(1 + (-3) = -2\)[/tex]
2. [tex]\((-1, 3)\)[/tex]: [tex]\(-1 + 3 = 2\)[/tex] (largest)
- When [tex]\(a=2\)[/tex], [tex]\(-3a = -6\)[/tex]:
1. [tex]\((1, -6)\)[/tex]: [tex]\(1 + (-6) = -5\)[/tex]
2. [tex]\((2, -3)\)[/tex]: [tex]\(2 + (-3) = -1\)[/tex]
3. [tex]\((-1, 6)\)[/tex]: [tex]\(-1 + 6 = 5\)[/tex] (largest)
- When [tex]\(a=3\)[/tex], [tex]\(-3a = -9\)[/tex]:
1. [tex]\((1, -9)\)[/tex]: [tex]\(1 + (-9) = -8\)[/tex]
2. [tex]\((3, -3)\)[/tex]: [tex]\(3 + (-3) = 0\)[/tex]
3. [tex]\((9, -1)\)[/tex]: [tex]\(9 + (-1) = 8\)[/tex] (largest)
4. [tex]\((-1, 9)\)[/tex]: [tex]\(-1 + 9 = 8\)[/tex] (also largest)
Taking this general pattern, it's observed clearly that:
The combination of pairs that maximizes the sum would often take forms with one positive and one negative component divided asymmetrically.
### Conclusion:
For general values of [tex]\(a\)[/tex]:
- The largest possible sum of integers that equates to the product [tex]\(-3a\)[/tex] is [tex]\(3a - 1\)[/tex], given the factors will try to balance out such as [tex]\( \left( \frac{x_1 y_1}{-3a}, \text{other terms balance with asymmetrical factors} \)[/tex].
Hence, the largest sum of two integers whose product is [tex]\(-3a\)[/tex] can be concluded to be as per the best possible set of pair factorization.
### Part 1: The product of two integers is -20
We need to find two integers whose product is [tex]\(-20\)[/tex] and determine the largest possible value of their sum.
To begin, list the factor pairs of [tex]\(-20\)[/tex]:
1. [tex]\((1, -20)\)[/tex]
2. [tex]\((-1, 20)\)[/tex]
3. [tex]\((2, -10)\)[/tex]
4. [tex]\((-2, 10)\)[/tex]
5. [tex]\((4, -5)\)[/tex]
6. [tex]\((-4, 5)\)[/tex]
Now, calculate the sum for each pair:
1. [tex]\(1 + (-20) = -19\)[/tex]
2. [tex]\(-1 + 20 = 19\)[/tex]
3. [tex]\(2 + (-10) = -8\)[/tex]
4. [tex]\(-2 + 10 = 8\)[/tex]
5. [tex]\(4 + (-5) = -1\)[/tex]
6. [tex]\(-4 + 5 = 1\)[/tex]
Among these, the largest sum is [tex]\(19\)[/tex], which occurs for the pair [tex]\((-1, 20)\)[/tex].
### Part 2: The product of two integers is [tex]\(-3a\)[/tex]
We need to find two integers whose product is [tex]\(-3a\)[/tex] and determine the largest possible value of their sum. Here, [tex]\(a\)[/tex] is a variable representing any positive integer.
For this more general case, let’s consider various factor pairs of [tex]\(-3a\)[/tex]:
Since [tex]\(-3a\)[/tex] has factor pairs [tex]\((p, \frac{-3a}{p})\)[/tex], the sum of these pairs would be:
[tex]\[ p + \frac{-3a}{p} \][/tex]
To maximize this sum, consider maximizing [tex]\(p\)[/tex] and minimizing [tex]\(\frac{-3a}{p}\)[/tex] in a balanced way.
#### Examples using specific values of [tex]\(a\)[/tex]:
- When [tex]\(a=1\)[/tex], [tex]\(-3a = -3\)[/tex]:
1. [tex]\((1, -3)\)[/tex]: [tex]\(1 + (-3) = -2\)[/tex]
2. [tex]\((-1, 3)\)[/tex]: [tex]\(-1 + 3 = 2\)[/tex] (largest)
- When [tex]\(a=2\)[/tex], [tex]\(-3a = -6\)[/tex]:
1. [tex]\((1, -6)\)[/tex]: [tex]\(1 + (-6) = -5\)[/tex]
2. [tex]\((2, -3)\)[/tex]: [tex]\(2 + (-3) = -1\)[/tex]
3. [tex]\((-1, 6)\)[/tex]: [tex]\(-1 + 6 = 5\)[/tex] (largest)
- When [tex]\(a=3\)[/tex], [tex]\(-3a = -9\)[/tex]:
1. [tex]\((1, -9)\)[/tex]: [tex]\(1 + (-9) = -8\)[/tex]
2. [tex]\((3, -3)\)[/tex]: [tex]\(3 + (-3) = 0\)[/tex]
3. [tex]\((9, -1)\)[/tex]: [tex]\(9 + (-1) = 8\)[/tex] (largest)
4. [tex]\((-1, 9)\)[/tex]: [tex]\(-1 + 9 = 8\)[/tex] (also largest)
Taking this general pattern, it's observed clearly that:
The combination of pairs that maximizes the sum would often take forms with one positive and one negative component divided asymmetrically.
### Conclusion:
For general values of [tex]\(a\)[/tex]:
- The largest possible sum of integers that equates to the product [tex]\(-3a\)[/tex] is [tex]\(3a - 1\)[/tex], given the factors will try to balance out such as [tex]\( \left( \frac{x_1 y_1}{-3a}, \text{other terms balance with asymmetrical factors} \)[/tex].
Hence, the largest sum of two integers whose product is [tex]\(-3a\)[/tex] can be concluded to be as per the best possible set of pair factorization.
