Exam 5: Number Theory

Find the least common multiple of the numbers in the group. -56, 48

For the following amicable pair, determine whether neither, one, or both of the members are happy, and whether the pair is a happy amicable pair. -79,750 and 88,730

Determine whether the statement is true or false. - $\text { If } \mathrm { n } \text { is a prime number, then the Mersenne number } 2 ^ { \mathrm { n } } - 1 \text { is also a prime number. }$

Find the least common multiple of the numbers in the group. -26,714, 3515

Find the greatest common factor of the numbers in the group. -16, 24, 28

Find the greatest common factor of the numbers in the group. -480, 27,000

There are 35 prime numbers smaller than 150.

Solve the problem. -Given the modulus n, the encrytion exponent e, and the plaintext M, use the RSA encrytion to find the ciphertext C. 161 7 8

Determine whether the statement is true or false. -If the least common multiple of $\mathrm { p }$ and $\mathrm { q }$ is smaller than $\mathrm { pq }$ , then $\mathrm { p }$ and $\mathrm { q }$ have a common factor other than 1 .

Use the Lucas sequence to answer the question. -What is the 20th term of the Lucas sequence?

Not all perfect numbers end in 6 or 28.

Determine whether the statement is true or false. - $\text { If } \mathrm { m } \text { divides } \mathrm { n } \text {, then } \mathrm { F } _ { \mathrm { m } } \text { is a factor of } \mathrm { F } _ { \mathrm { n } } \cdot \mathrm { F } _ { \mathrm { n } } \text { represents the } \mathrm { nth } \text { term of the Fibonacci sequence. }$

All composite numbers are also abundant numbers.

If a number is divisible by both 3 and 9 then it is divisible by 27.

Use inductive reasoning to find the next equation in the pattern. -2 - 1 = 1 3 - 2 = 1 5 - 3 = 2 8 - 5 = 3

Use the formula indicated to determine the prime number generated for the given value of n. - $p = n ^ { 2 } - n + 41 ; n = 13$

Give the prime factorization of the number. Use exponents when possible. -154

If a natural number is divisible by 2, then it must also be divisible by 10.

Use divisibility tests to decide whether the first number is divisible by the second. -139,243,105; 11 [Note that a divisibility test for 11 is as follows: Starting at the left of the number, add together every other digit. Add together the remaining digits. Subtract the smaller of the two sums from the larger. If the final number obtained is divisible by 11, Then so is the original number. If the final number obtained is not divisible by 11, then neither is the Original number.]

Determine whether the statement is true or false. -The least common multiple of p and q cannot be larger than pq.