Let's address each part of the question carefully.
### Part (c):
We need to determine if [tex]\( m^2 + 2mn + n^2 \)[/tex] is always a composite number when [tex]\( m \)[/tex] and [tex]\( n \)[/tex] are positive integers.
Step-by-Step Solution:
1. Expression Understanding: The given expression is [tex]\( m^2 + 2mn + n^2 \)[/tex].
2. Simplification: Notice that this can be rewritten using the algebraic identity for the square of a binomial:
[tex]\[
m^2 + 2mn + n^2 = (m + n)^2
\][/tex]
3. Nature of [tex]\( (m + n)^2 \)[/tex]: Since [tex]\( m \)[/tex] and [tex]\( n \)[/tex] are positive integers, the sum [tex]\( m + n \)[/tex] is also a positive integer. Consequently, [tex]\( (m + n)^2 \)[/tex] is the square of a positive integer.
4. Composite Number Definition: A composite number is defined as a positive integer that has at least one positive divisor other than one and itself. Since [tex]\( m \)[/tex] and [tex]\( n \)[/tex] are positive, [tex]\( (m + n)^2 \)[/tex] is a positive integer that is the product of [tex]\( (m + n) \times (m + n) \)[/tex].
5. Conclusion: A number squared ([tex]\( (m+n)^2 \)[/tex]) is always greater than 1 and can be factored into two equal positive integers both greater than 1, making it composite.
Hence, [tex]\( m^2 + 2mn + n^2 \)[/tex] is always composite when [tex]\( m \)[/tex] and [tex]\( n \)[/tex] are positive integers.
### Part (d):
We need to determine if [tex]\( m^2 - n^2 \)[/tex] is always a composite number under the condition [tex]\( m \ge n + 2 \ge 3 \)[/tex].
Step-by-Step Solution:
1. Expression Understanding: The given expression is [tex]\( m^2 - n^2 \)[/tex].
2. Simplification: Notice that this can be rewritten using the difference of squares formula:
[tex]\[
m^2 - n^2 = (m + n)(m - n)
\][/tex]
3. Conditions: We are given that [tex]\( m \ge n + 2 \ge 3 \)[/tex].
4. Nature of [tex]\( m \)[/tex] and [tex]\( n \)[/tex]: Because [tex]\( n \ge 1 \)[/tex] (since [tex]\( n + 2 \ge 3 \)[/tex] implies [tex]\( n \ge 1 \)[/tex]) and [tex]\( m \ge n + 2 \)[/tex], we know that both [tex]\( m \)[/tex] and [tex]\( n \)[/tex] are positive integers.
5. Analysis of [tex]\( (m + n)(m - n) \)[/tex]:
- Since [tex]\( m \ge n + 2 \)[/tex], [tex]\( m - n \ge 2 \)[/tex], making [tex]\( m - n \)[/tex] a positive integer greater than 1.
- [tex]\( m + n \)[/tex] is also greater than [tex]\( n + 2 + n \ge n + 2 \ge 3 \)[/tex], resulting in [tex]\( m + n \)[/tex] being a positive integer greater than 1.
6. Conclusion: The product [tex]\( (m + n)(m - n) \)[/tex] will be a product of two positive integers greater than 1, which means it is a composite number.
Hence, [tex]\( m^2 - n^2 \)[/tex] is always composite under the condition [tex]\( m \ge n + 2 \ge 3 \)[/tex].
