Answer :
To address the problem, let's analyze the sums of pairs of even numbers and formulate some conjectures based on the results.
Given a pattern of sums:
[tex]\[ \begin{array}{l} 4 + 8 = 12 \\ 6 + 20 = 26 \\ 16 + 32 = 48 \\ 24 + 48 = 72 \\ 36 + 52 = 88 \\ \end{array} \][/tex]
We will check the sums and the properties they exhibit to form accurate conjectures.
### Step-by-Step Analysis:
#### 1. Sum of Two Even Numbers:
- Recall that the sum of two even numbers is:
[tex]\[ \text{Sum} = \text{Even Number 1} + \text{Even Number 2} \][/tex]
#### 2. Calculating Each Sum:
- Sum of [tex]\(4\)[/tex] and [tex]\(8\)[/tex]: [tex]\(4 + 8 = 12\)[/tex]
- Sum of [tex]\(6\)[/tex] and [tex]\(20\)[/tex]: [tex]\(6 + 20 = 26\)[/tex]
- Sum of [tex]\(16\)[/tex] and [tex]\(32\)[/tex]: [tex]\(16 + 32 = 48\)[/tex]
- Sum of [tex]\(24\)[/tex] and [tex]\(48\)[/tex]: [tex]\(24 + 48 = 72\)[/tex]
- Sum of [tex]\(36\)[/tex] and [tex]\(52\)[/tex]: [tex]\(36 + 52 = 88\)[/tex]
We obtain the sums: [tex]\(12, 26, 48, 72, 88\)[/tex].
#### 3. Conjectures Based on Results:
Conjecture 1: Sum of Two Even Numbers is Always Even
- Observing the sums: [tex]\(12, 26, 48, 72, 88\)[/tex]
- Each of these values is even.
Conclusion:
[tex]\(\boxed{\text{The sum of two even numbers is always even.}}\)[/tex]
Conjecture 2: Sum of Two Even Numbers is Divisible by 3
- Check if each sum is divisible by 3:
- [tex]\(12 \div 3 = 4\)[/tex]: Divisible
- [tex]\(26 \div 3 \approx 8.67\)[/tex]: Not Divisible
- [tex]\(48 \div 3 = 16\)[/tex]: Divisible
- [tex]\(72 \div 3 = 24\)[/tex]: Divisible
- [tex]\(88 \div 3 \approx 29.33\)[/tex]: Not Divisible
Conclusion:
[tex]\(\boxed{\text{The sum of two even numbers is not always divisible by 3.}}\)[/tex]
### Final Conjectures:
1. The sum of two even numbers is always an even number.
2. The sum of two even numbers is not always divisible by 3.
These conclusions are based on the pattern analysis of the given examples.
Given a pattern of sums:
[tex]\[ \begin{array}{l} 4 + 8 = 12 \\ 6 + 20 = 26 \\ 16 + 32 = 48 \\ 24 + 48 = 72 \\ 36 + 52 = 88 \\ \end{array} \][/tex]
We will check the sums and the properties they exhibit to form accurate conjectures.
### Step-by-Step Analysis:
#### 1. Sum of Two Even Numbers:
- Recall that the sum of two even numbers is:
[tex]\[ \text{Sum} = \text{Even Number 1} + \text{Even Number 2} \][/tex]
#### 2. Calculating Each Sum:
- Sum of [tex]\(4\)[/tex] and [tex]\(8\)[/tex]: [tex]\(4 + 8 = 12\)[/tex]
- Sum of [tex]\(6\)[/tex] and [tex]\(20\)[/tex]: [tex]\(6 + 20 = 26\)[/tex]
- Sum of [tex]\(16\)[/tex] and [tex]\(32\)[/tex]: [tex]\(16 + 32 = 48\)[/tex]
- Sum of [tex]\(24\)[/tex] and [tex]\(48\)[/tex]: [tex]\(24 + 48 = 72\)[/tex]
- Sum of [tex]\(36\)[/tex] and [tex]\(52\)[/tex]: [tex]\(36 + 52 = 88\)[/tex]
We obtain the sums: [tex]\(12, 26, 48, 72, 88\)[/tex].
#### 3. Conjectures Based on Results:
Conjecture 1: Sum of Two Even Numbers is Always Even
- Observing the sums: [tex]\(12, 26, 48, 72, 88\)[/tex]
- Each of these values is even.
Conclusion:
[tex]\(\boxed{\text{The sum of two even numbers is always even.}}\)[/tex]
Conjecture 2: Sum of Two Even Numbers is Divisible by 3
- Check if each sum is divisible by 3:
- [tex]\(12 \div 3 = 4\)[/tex]: Divisible
- [tex]\(26 \div 3 \approx 8.67\)[/tex]: Not Divisible
- [tex]\(48 \div 3 = 16\)[/tex]: Divisible
- [tex]\(72 \div 3 = 24\)[/tex]: Divisible
- [tex]\(88 \div 3 \approx 29.33\)[/tex]: Not Divisible
Conclusion:
[tex]\(\boxed{\text{The sum of two even numbers is not always divisible by 3.}}\)[/tex]
### Final Conjectures:
1. The sum of two even numbers is always an even number.
2. The sum of two even numbers is not always divisible by 3.
These conclusions are based on the pattern analysis of the given examples.
