Exam 5: Number Theory

Answer the question. -Mark has 153 hot dogs and 261 hot dog buns. He wants to put the same number of hot dogs and hot dog buns on each tray. What is the greatest number of trays Mark can use to accomplish this?

Solve the problem. -Euler conjectured that the formula $\mathrm { n } ^ { 2 } - \mathrm { n } + 41$ would generate prime numbers for whole numbers $\mathrm { n }$ . The formula generates prime numbers for $\mathrm { n }$ up to 40 and fails at $\mathrm { n } = 41$ . Evaluate Euler's formula for $n = 42$ and determine whether this is a prime number. If it is not a prime number, give its prime factors.

Answer the question. -At Northwest High School, there are 621 students in the Junior Class and 897 students in the Senior Class. To let the juniors work with more experienced students, the teachers want to assign the Students to committees with the same number of juniors in each committee and the same number Of seniors in each committee. (For example, there might be 2 juniors and 3 seniors in every Committee). What is the largest number of committees that can be formed?

Solve the problem. -Apply the RSA Scheme to find each missing value. \ell 7 13

Solve the problem. -Given the prime factors p and q, the encrytion exponent e, and the ciphertext C, apply the RSA algorithm to find the plaintext message. 23 17 11 114

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes. -32 and 39

Find the greatest common factor of the numbers in the group. -84, 378

Use the formula indicated to determine the prime number generated for the given value of n. - $p = n ^ { 2 } - 79 n + 1,601 ; n = 43$

Give the prime factorization of the number. Use exponents when possible. -936

Determine all values for the digit x that make the first number divisible by the second number. If none exist, so state. -(43 \times 1 \text { is divisible by } 3 \text {. }\)

Write the number as the sum of two primes. There may be more than one way to do this. -44

Determine whether the number is abundant or deficient. -95

Find the greatest common factor of the numbers in the group. -240, 10,000, 11,250

Twin primes are prime numbers whose difference is a multiple of 4.

Answer the question. -Several different bus routes stop at the corner of Second St. and Lincoln Ave. A Wilkenson bus arrives every 21 minutes and a Harris Road bus arrives every 15 minutes. If both buses arrive at the Stop at 5:07 AM, when will they again arrive at the same time?

If n is a natural number and 20|n, then 10|n.

Find the residue. -484 (mod 15)

Solve the problem relating to the Fibonacci sequence. - $\mathrm { F } _ { 28 } = 317,811 , \mathrm {~F} _ { 30 } = 832,040$ Find $\mathrm { F } _ { 29 }$ .

There is no largest prime number.

Determine whether the number is abundant or deficient. -8