Exam 5: Number Theory

Use the formula indicated to determine the prime number generated for the given value of n. - $p = n ^ { 2 } - n + 41 ; n = 8$

Use the formula indicated to determine the prime number generated for the given value of n. - $p = n ^ { 2 } - n + 41 ; n = 28$

Solve the problem. -Given that the following statement is true, explain how you would use this fact to find two distinct factors of the Mersenne number $\mathrm { M } _ { 14 }$ . Then calculate those two factors. "If $\mathrm { p }$ is a prime factor of $\mathrm { n }$ , then $2 \mathrm { p } - 1$ is a factor of the Mersenne number $2 ^ { \mathrm { n } } - 1$ ."

Determine whether the statement is true or false. - $\text { The number } 2 ^ { 7 } \left( 2 ^ { 8 } - 1 \right) \text { is perfect. }$

Determine whether the statement is true or false. -The greatest common factor of a group of natural numbers is the largest natural number that is a factor of all the numbers in a group.

Determine whether the statement is true or false. - $\text { Escott's formula } n ^ { 2 } - 79 n + 1601 \text { first fails at } n = 80$

Find the least common multiple of the numbers in the group. -135, 28, 150

Determine whether the statement is true or false. -A Pythagorean triple can be obtained as follows: Choose any four successive terms of the Fibonacci sequence. Multiply the first and fourth. Double the product of the second and third. Add the squares of the second and third. The three numbers obtained will always form a Pythagorean triple.

Solve the problem. -Escott used the formula $n ^ { 2 } - 79 n + 1601$ to generate prime numbers for whole numbers $n$ . The formula fails to produce a prime for $\mathrm { n } = 80$ . Evaluate Escott's formula for $\mathrm { n } = 81$ and determine whether this is a prime number. If it is not a prime number, give its prime factors.

Answer the question. -A brick layer is hired to build three walls of equal length. He has three lengths of brick, 9 inches, 27 inches, and 21 inches. He plans to build one wall out of each type. What is the shortest length of Wall possible?

Answer the question. -Bob's frog travels 7 inches per jump, Kim's frog travels 9 inches and Jack's frog travels 13 inches. If the three frogs start off side-by-side, what is the smallest distance they must all travel before they Are side-by-side again?

Give the prime factorization of the number. Use exponents when possible. -28

Solve the problem. -Fermat conjectured that the formula $2 ^ { 2 ^ { n } } + 1$ would always produce a prime for any whole number $\mathrm { n }$ . The sixth Fermat number is $2 ^ { 2 ^ { 5 } } + 1$ . Evaluate this number. In seeking possible prime factors of this Fermat number, what is the largest potential prime factor that one would have to try?

Determine all values for the digit x that make the first number divisible by the second number. If none exist, so state. -533x is divisible by 11.

Any composite number in the range 2 to $n$ must be a multiple of some prime number less than or equal to $\sqrt { n }$ .

Determine whether the statement is true or false. - $\text { The number } 2 ^ { 13 } - 1 \text { is an example of a Mersenne prime. }$

Determine whether the statement is true or false. - $\text { Euler's formula } \mathrm { n } ^ { 2 } - \mathrm { n } + 41 \text { generates primes for } \mathrm { n } \text { up to } 46 \text { and fails at } \mathrm { n } = 47$

Solve the problem relating to the Fibonacci sequence. -If a 13-inch wide rectangle is to approach the golden ratio, what should its length be?

Determine all values for the digit x that make the first number divisible by the second number. If none exist, so state. -417,30x is divisible by 8 but not 16.

Use the Lucas sequence to answer the question. -What is the 30th term of the Lucas sequence?