Exam 1 Review:

The following is a list of terms or phrases that you need to be able to explain their meaning and give examples.

numerical partitions

proper divisor

prime number

composite number

abundant number

deficient number

perfect number

greatest common divisor

relatively prime numbers

multiplication modulo n

divisor of zero

multiplicative inverse of a number

1.                           Find all the partitions of 12 using 3 or fewer numbers. The number of partitions found is p(12, 3) + p(12,2) + p(12,1). Give its value.

Find all the partitions of 12 using numbers less than or equal to 3. The number of partitions found is p(12)3. Give its value.

What do you observe about these two results?

2.           The number of partitions of n into k parts, p(n, k), satisfies the recurrence relation

        p(n, k) = p(n - 1,k - 1) + p(n - k, k).

If you want to know p(14, 5) what do you need to know to use the formula?

3.                           Use the Sieve of Eratosthenes to find the prime numbers less than 50. Explain why 7 is the largest prime number needed to find the primes less than 50.

4.           Show that 30 is an abundant number by using the definition.

5.           Explain why the number 74070 is abundant without finding all of its divisors.

6.           All known perfect numbers satisfy the formula

        2n – 1 ´ (2n – 1).

Show that 28 fits this formula. If we let n = 15, will the result be a perfect number? Explain.

7.           Find the GCD and LCM of each pair of numbers.  Explain your method. (Hint: you do not need the Euclidean Algorithm.)

        (7, 15)             (288, 36)        

8.           Use the Euclidean Algorithm to find the GCD of 38240 and 14950.

9.                           Compute 5 ´12 9; 5 ´13 9; 5 ´14 9

10.                     Find all the numbers less than 24 that are relatively prime to 24.  Find all the numbers less than 24 that are divisors of zero.

11.                     Create a multiplication table for multiplication modulo 9.

12.                     Use your multiplication table in #11 to compute 7 ÷9 2 , 2 ÷9 5.

13.                     Use your multiplication table in #11 to explain why 4 ÷9 6 is undefined.

14.                     Use your multiplication table in #11 to find the multiplicative inverse modulo 9 of 2. Repeat for 7. Explain why 6 does not have a multiplicative inverse modulo 9.

15.                     List two important ways that a multiplication modulo n table changes when the divisors of zero are removed.  n is assumed to be a composite number.