Number Theory Review 1

    Here is a list of 30 problems that will help a number theory student understanding their working knowledge and skill level for elementary number theory.

(1) Use mathematical induction to prove that whenever is a positive integer.

(2) Use mathematical induction to prove that for every positive integer

(3) Show that any amount of postage that is an integer number of cents greater than 53 cents can be formed using just 7-cent and 10-cent stamps.

(4) Use mathematical induction to prove that for

(5) Use mathematical induction to prove that for

(6) Find the Fibonacci number

(7) Find the Fibonacci number

(8) Show that whenever is a positive integer.

(9) Show that whenever is a positive integer.

(10) Show that whenever is a positive integer.

(11) Find the quotient and remainder in the division algorithm, (a) with divisor 17 and dividend (b) with divisor 17 and dividend

(12) What can you conclude if and are nonzero integers such that and

(13) Show that if is an integer, then divides

(14) Show that the product of every two integers of the form is of the form

(15) Use mathematical induction to show that the sum of the cubes of three consecutive positive integers is divisible by 9.

(16) Determine which of the following integers are prime:

    (a) 201   (b) 213   (c) 221   (d) 203   (e) 207   (f) 211

(17) Find all primes that are the difference of the fourth powers of two integers.

(18)  Show that no integer of the form is a prime, other than

(19) Find the smallest prime in the arithmetic progression where

    (a)    (b)    (c)
    

(20) Show that is prime for all integers with Show that it is composite for   

(21) Find the greatest common divisor of each of the following pairs of integers:

    (a)    (b) 0,100   (c)    (d)    (e) 100, 121   (f) 1001, 289

(22) Let b a positive integer. What is the greatest common divisor of and

(23) Show that if and are integers with then or 2.

(24) Show that if and are integers with then

(25) Show that and are relatively prime for all integers

(26) Use the Euclidean Algorithm to find the greatest common divisor and then write this as a linear combination of these integers.

(27) Use the Euclidean Algorithm to find the greatest common divisor and then write this as a linear combination of these integers.

(28) Use the Euclidean Algorithm to find the greatest common divisor and then write this as a linear combination of these integers.

(29) Find the prime factorization of

(30) Show that if and are positive integers and then

(31) Show that if and then

(32) Find the least common multiple of each of the following pairs of integers:

(33) Show that if and are integers, then if and only if and

• print this page
• pages linking here
• related pages