Alright, let's break it down step by step.
### For the Product of -20:
1. Identify Pairs with Product -20:
- We need pairs of integers (x, y) such that their product is -20, i.e., [tex]\( x \times y = -20 \)[/tex].
- The pairs of integers that satisfy this equation are:
- [tex]\( (1, -20) \)[/tex]
- [tex]\( (2, -10) \)[/tex]
- [tex]\( (4, -5) \)[/tex]
- [tex]\( (5, -4) \)[/tex]
- [tex]\( (10, -2) \)[/tex]
- [tex]\( (20, -1) \)[/tex]
2. Calculate the Sum of Each Pair:
- [tex]\( 1 + (-20) = -19 \)[/tex]
- [tex]\( 2 + (-10) = -8 \)[/tex]
- [tex]\( 4 + (-5) = -1 \)[/tex]
- [tex]\( 5 + (-4) = 1 \)[/tex]
- [tex]\( 10 + (-2) = 8 \)[/tex]
- [tex]\( 20 + (-1) = 19 \)[/tex]
3. Find the Maximum Sum:
- Among the sums [tex]\([-19, -8, -1, 1, 8, 19]\)[/tex], the largest value is 19.
Therefore, the largest possible sum of two integers whose product is -20 is 19.
### For the Product of -30:
1. Identify Pairs with Product -30:
- We need pairs of integers (x, y) such that their product is -30, i.e., [tex]\( x \times y = -30 \)[/tex].
- The pairs of integers that satisfy this equation are:
- [tex]\( (1, -30) \)[/tex]
- [tex]\( (2, -15) \)[/tex]
- [tex]\( (3, -10) \)[/tex]
- [tex]\( (5, -6) \)[/tex]
- [tex]\( (6, -5) \)[/tex]
- [tex]\( (10, -3) \)[/tex]
- [tex]\( (15, -2) \)[/tex]
- [tex]\( (30, -1) \)[/tex]
2. Calculate the Sum of Each Pair:
- [tex]\( 1 + (-30) = -29 \)[/tex]
- [tex]\( 2 + (-15) = -13 \)[/tex]
- [tex]\( 3 + (-10) = -7 \)[/tex]
- [tex]\( 5 + (-6) = -1 \)[/tex]
- [tex]\( 6 + (-5) = 1 \)[/tex]
- [tex]\( 10 + (-3) = 7 \)[/tex]
- [tex]\( 15 + (-2) = 13 \)[/tex]
- [tex]\( 30 + (-1) = 29 \)[/tex]
3. Find the Maximum Sum:
- Among the sums [tex]\([-29, -13, -7, -1, 1, 7, 13, 29]\)[/tex], the largest value is 29.
Therefore, the largest possible sum of two integers whose product is -30 is 29.
To summarize:
1. The largest possible value of the sum of two integers whose product is -20 is 19.
2. The largest possible value of the sum of two integers whose product is -30 is 29.
