Number Theory Review 3

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

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

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

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

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

(6) Find the day of the week for: March 30, 1867 (U.S. buys Alaska from Russia.)

(7) Find the day of the week for: July 4, 1776 (U.S. Declaration of Independence.)

(8) Find the day of the week for: July 20, 1969 (First man on the moon)

(9) Show that every year in the Gregorian calendar includes at least one Friday the 13th.

(10) Show that is divisible by 11 by grouping together pairs of inverses modulo 11 that occur in

(11) What is the remainder when is divided by 437?

(12) Using Fermat's little theorem, find the least positive residue of modulo 17.

(13)  Using Fermat's little theorem, solve the linear congruence and

(14) Show that if is prime then

(15) Find a reduced residue system modulo

(16) Find a reduced residue system modulo

(17) Show that if and are positive integers with then

(18) Find the last digit in the decimal expansion of

(19) Use Euler's theorem to find the least positive residue of modulo 35.

(20) Find the value of the Euler phi-function for the integer 100.

(21) Find the value of the Euler phi-function for the integer 256.

(22) Find the value of the Euler phi-function for the integer

(23) Show that

(24) Find all positive integers such that

(25) Show that if and are positive integers, then   

(26) Find all of the quadratic residues of the integer 8.

(27) Evaluate the Legendre symbol using Euler's criterion.

(28) Evaluate the Legendre symbol using Gauss's lemma.

(29) Find all solutions of the quadratic congruence

(30) Find all solutions of the quadratic congruence

(31) Evaluate the Legendre symbol

(32) Evaluate the Legendre symbol   

(33) Evaluate the Legendre symbol   

(34) Using the law of quadratic reciprocity, show that if is an odd prime. Then

(35) Using the law of quadratic reciprocity, show that if is an odd prime. Then

(36) Find a congruence describing all primes for which 5 is a quadratic residue.

(37) Find the rational number, expressed in lowest terms, represented by the finite continued fraction

(38) Find the rational number, expressed in lowest terms, represented by the finite continued fraction

(39) Find the simple continued fraction expansion, not terminating with the partial quotient of 1, for the rational number

(40) Find the simple continued fraction expansion, not terminating with the partial quotient of 1, for the rational number

(41) Find the simple continued fraction expansion, not terminating with the partial quotient of 1, for the rational number

(42) Let denote the -th Fibonacci number. Find the simple continued fraction, terminating with the partial quotient of 1, of , where is a positive integer.   

(43) Show that if the simple continued fraction expression of the rational number is then the simple continued fraction expression of is

(44) Find the simple continued fraction of the real number .

(45) Find the simple continued fraction of the real number .

(46) Find the first five partial quotients of the simple continued fractions of the real number .

(47) Find the first five partial quotients of the simple continued fractions of the real number .

(48) Find the best rational approximation to with a denominator less than or equal to 100,000.

(49) Find the best rational approximation to with a denominator less than or equal to 536.

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