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10) Make a table of values for the rule x2 + x + 11 when x is an integer from 1 to 8. Make a conjecture about the type of number generated by the rule. Continue your table. What value of x generates a counterexample?

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We have been given a rule
x^(2)+x+11 and we are required to prepare a table of values using this rule and values of x from 1 to 8.

For x=1, we get:
1^(2)+1+11=13

For x=2, we get:
2^(2)+2+11=17

For x=3, we get:
3^(2)+3+11=23

For x=4, we get:
4^(2)+4+11=31

For x=5, we get:
5^(2)+5+11=41

For x=6, we get:
6^(2)+6+11=53

For x=7, we get:
7^(2)+7+11=67

For x=8, we get:
8^(2)+8+11=83

We can see that all these values are prime numbers. So, we can make a conjecture that the given rule produces prime numbers.

Now we need to find a counter example, that is, a value of x that produces a non-prime number, that is, a composite number.

Let us calculate the value of the given function at x=10. We get the value of the function as:

For x=10, we get:
10^(2)+10+11=121

We know that 121 is a composite number as it is divisible by 11.

Therefore, x=10 generates a counter example.


User David Heffernan
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