3. Kate was given the following number pattern by her teacher. 22+1=5 42+1=17 62+1=37 102+1=101 14²+1=197 Kate made the following conjecture based solely on the number patterns shown. "When an even number is squared and one is added to it, the resulting number is a prime number". In the examples above 5, 17, 37, 101, and 197 are prime numbers because the only factors of these numbers are 1 and the number itself. a) Is Kate's conjecture True or False, based solely on the numbers shown? Use your understanding of even numbers and prime numbers to explain your answer. b) Determine a counterexample to the pattern above to show that Kate's conjecture is false.
3. Kate was given the following number pattern by her teacher. 22+1=5 42+1=17 62+1=37 102+1=101 14²+1=197 Kate made the following conjecture based solely on the number patterns shown. "When an even number is squared and one is added to it, the resulting number is a prime number". In the examples above 5, 17, 37, 101, and 197 are prime numbers because the only factors of these numbers are 1 and the number itself. a) Is Kate's conjecture True or False, based solely on the numbers shown? Use your understanding of even numbers and prime numbers to explain your answer. b) Determine a counterexample to the pattern above to show that Kate's conjecture is false.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:3. Kate was given the following number pattern by her teacher.
22+1=5
42+1=17
62+1=37
102+1=101
14²+1=197
Kate made the following conjecture based solely on the number patterns shown.
"When an even number is squared and one is added to it, the resulting number is a prime
number". In the examples above 5, 17, 37, 101, and 197 are prime numbers because the only
factors of these numbers are 1 and the number itself.
a) Is Kate's conjecture True or False, based solely on the numbers shown? Use your
understanding of even numbers and prime numbers to explain your answer.
b) Determine a counterexample to the pattern above to show that Kate's conjecture is false.
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