Let's analyze each part and find appropriate counterexamples and the correct choice for the most general criteria.
### Part a
We need to find a counterexample for the statement:
[tex]\[ \frac{n+5}{5} = n+1 \][/tex]
where [tex]\( n \)[/tex] is a natural number.
To find a counterexample, let's test this equation with a natural number [tex]\( n \)[/tex]. For instance, let's take [tex]\( n = 1 \)[/tex].
1. Compute the left-hand side:
[tex]\[ \frac{1+5}{5} = \frac{6}{5} = 1.2 \][/tex]
2. Compute the right-hand side:
[tex]\[ 1+1 = 2 \][/tex]
Clearly, [tex]\( \frac{6}{5} \neq 2 \)[/tex]. Thus, [tex]\( n = 1 \)[/tex] is a counterexample.
Now, let's determine the most general criteria. The choices given are:
A. Any natural number [tex]\( n \)[/tex] that is greater than 5 can be used as a counterexample.
B. Any natural number [tex]\( n \)[/tex] that is not equal to 5 can be used as a counterexample.
C. Any natural number [tex]\( n \)[/tex] can be used as a counterexample.
D. Any natural number [tex]\( n \)[/tex] that is less than 5 can be used as a counterexample.
Since the equation does not hold for any natural number except possibly [tex]\( n = 5 \)[/tex], the correct choice is B. Any natural number [tex]\( n \)[/tex] that is not equal to 5 can be used as a counterexample.
### Part b
We need to find a counterexample for the statement:
[tex]\[ (n+4)^2 = n^2 + 16 \][/tex]
where [tex]\( n \)[/tex] is a natural number.
To find a counterexample, let's test this equation with a natural number [tex]\( n \)[/tex]. For instance, let's take [tex]\( n = 2 \)[/tex].
1. Compute the left-hand side:
[tex]\[ (2+4)^2 = 6^2 = 36 \][/tex]
2. Compute the right-hand side:
[tex]\[ 2^2 + 16 = 4 + 16 = 20 \][/tex]
Clearly, [tex]\( 36 \neq 20 \)[/tex]. Thus, [tex]\( n = 2 \)[/tex] is a counterexample.
### Summary
a. Choice B (Any natural number [tex]\( n \)[/tex] that is not equal to 5 can be used as a counterexample) is the most general criteria.
b. A counterexample for the statement [tex]\((n+4)^2 = n^2 + 16\)[/tex] is [tex]\( n = 2 \)[/tex].
Therefore, the solution is:
1. (True, as [tex]\( n=1 \)[/tex]) for [tex]\(\frac{n+5}{5} = n+1\)[/tex].
2. (True, as [tex]\( n=2 \)[/tex]) for [tex]\((n+4)^2 = n^2 + 16\)[/tex].
3. The most general choice for part a is [tex]\( B \)[/tex].
