MTH 3527 Number Theory
Some Extra Problems
Name:
1. Which primes can be written as sums of two squares?
2. Which integers can be written as sums of two squares?
3. Consider the prime p = 61. Find an integer x such that 2 x 60 and
x2 (1) (mod 61).
4. Consi
MTH 3527 Number Theory Quiz 10
(Some problems that might be on the quiz and some solutions.)
1. Euler -function. Desribe all integers n such that:
(a) (n) = 2
Solution: n = 4, 3, 6 since (22 ) = 2(2 1) = 2, (3) = 3 1 = 2, (6) = (2)(3) =
12=2
(b) (n) = 4
S
MTH 3527 Number Theory
Quiz 10
Name:
1. Find all integers n 100 so that 5|(n), where (n) is the Euler function.
Solution Step 1. Find all primes p such that 5 | (pk ).
Let p be prime. Then (pk ) = pk1 (p 1).
If 5|(p) then, one possibility is: p = 5 and k
MTH 3527 Number Theory
Quiz 9 (Only some solutions)
Name:
When using any of the theorems, propositions, statements or claims which were already
proved, write the complete statement clearly (not just refering to them by name)!
For example: instead of writi
MTH 3527 Number Theory
Quiz 6
Name:
When using any theorem, make sure that you state it precisely!
1. Determine all positive integers n such that (n) = 40.
Solution (n) = 40 = 23 5.
n = 41, 2 41, 5 11, 2 5 11, 22 3 11, 23 11, 3 52 , 2 3 52 , 22 52 ,
2. (a
MTH 3527 Number Theory
Quiz 6 (Practice)
Name:
1. Consider all positive integers (mod7). How many primes does each equivalence class have?
Explain your statement).
2. Prove that there are innitely many primes congruent to 5 modulo 6.
3. For which values o
MTH 3527 Number Theory
Quiz 7
1. Find 335 (mod 561) using method of successive squaring.
2. Find x such that x35 3(mod 89).
1
Name:
3. Find x such that x35 3(mod 65).
4. Find x such that x35 3(mod 169).
2
MTH 3527 Number Theory
Quiz 6
Name:
When using any theorem, make sure that you state it precisely!
1. Determine all positive integers n such that (n) = 40.
2. (a) Prove that there are no primes p such that p 10 (mod 18).
(b) Prove that there is exactly on
MTH 3527 Number Theory
Quiz 5
Name:
1. Using Chinese Remainder Theorem nd x such that
x 27(mod 31), x 50(mod 199) and 0 x < (31)(199) = 6169.
State the theorem, clearly indicate what you are doing at each step, and check your solution.
1
2. You are told t
MTH 3527 Number Theory
Quiz 4
Name:
For this quiz, you always have to state precisely what you are using!
1. Find a number a, such that 0 < a < 97 and a 43961 (mod 97).
2. Solve the following equation x333 5(mod 11). Find all non-congruent solutions.
3. S
MTH 3527 Number Theory
Quiz 3
Name:
1. Find all incongruent solutions to the following linear congruence:
2. Give a proof by induction: 1 + a + a2 + + an =
1
1an+1
.
1a
66x 100 (mod 121) .
3. For each of the following linear congruences, decide how many n
MTH 3527 Number Theory Quiz 2
1. Let a, b, c be positive integers. Either prove or nd a counterexample:
(a) If 6|c and 10|c then 60|c
(b) If a|c and b|c and gdc(a, b) = 1 then ab|c.
2. Use Euclidean algorithm to:
(a) Find gcd(13243, 10336) = g.
(b) Find x
MTH 3527 Number Theory
Quiz 1
Show all work - Circle Answers
Name
1. List rst ve even triangular numbers.
2. (a) Let a be an odd integer. Prove that a2 = 8Tk + 1, where Tk is a triangular integer.
(b) Find k so that 112 = 8Tk + 1, where Tk is a triangular
MTH 3527 Number Theory
Quiz 1(Practice)
Show all work - Circle Answers
Name
1. (a) List rst ve odd triangular numbers.
(b) List rst ve even triangular numbers.
2. (a) Let Tn be a triangular number. Prove that 8Tn + 1 is a square of an odd integer.
(b) Acc