Old Quizzes

Quiz 8 (due 3/28)

Fact: 3301 and 3307 are prime.
1. What is phi(3301) ?
2. Is there a primitive root for p = 3307 ? (If there is one, you don't need to find a value.)
3. Is there an x such that (x^2 - 2) is a multiple of 3307 ? (If there is one, you don't need to find a value.)

Quiz 7 (due 3/21)

1. Use Fermat's Little Theorem to determine the value of 19^30 (that is, 19 to the power 30) ( mod 31 ).
2. Use Fermat's Little Theorem (or any method) to determine the value of 9^101 ( mod 11 ).
3. Use Wilson's Theorem to determine the value of
1^2 times 2^2 times 3^2 times ... times (p-1)^2 (mod p)
if p is prime.

Quiz 6 (due 3/7)

1. Find a number x such that 103x is congruent to 4 (mod 119). Hint: 52 times 103 is congruent to 1 mod 119.
2. Find ALL of the solutions to: 25x congruent to 75 (mod 125).
3. (no points) Are any of the solutions to problem 2 perfect squares?
4. What is 5! (mod 6) ?
5. What is 9! (mod 10) ?

Quiz 5 (due 2/28)

1. What is the value of 10000000003, reduced mod 4 ?
2. What is the value of 23456780007, reduced mod 8 ?
3. If x is an integer, what are the possible values of x^2 (that's "x squared"), reduced mod 4 ?
4. If x is an integer, what are the possible values of x^2 reduced mod 8 ?
5. Can you find integers a and b such that a^2 + b^2 = 10000000003 ? Why not?
6. Can you find integers a, b, and c such that a^2 + b^2 + c^2 = 23456780007 ? Why not?
Not part of quiz:Can you find integers a, b, c, and d such that a^2 + b^2 + c^2 + d^2 = 2718281828 ?

Quiz 4 (due 2/14)

1. What is ( 3 * 5^3 * 7^2 , 2^2 * 5 * 7 * 11 ) ? [You don't need to multiply any of the answers out. Guide to typography: * means times. ^ means exponentiation, so that "7^2" means "7 with exponent 2" or "7 squared."]
2. What is [ 3 * 5^3 * 7^2 , 2^2 * 5 * 7 * 11 ] ?
3. If (323, 2975) = 17, what is [323, 2975] ?
4. Are there infinitely many primes of the form 6k+4, where k is an integer? Why or why not?
5. Are there infinitely many primes of the form 4k+7, where k is an integer? Why or why not? (a line or two is enough; a full proof isn't needed on problem 4 or problem 5.)

Quiz 3 (due 2/7)

1. What is the greatest common divisor of 454185 and 918463 ?
2. Can you express 3 as a linear combination of 81 and 41 ?
3. Notice that 323 divides 678300 (which is 2975 times 228) but 323 does not divide 2975. What can you conclude?

Quiz 2 (due 1/31)

1. Express the decimal number 99 in binary notation.
2. Express the binary number 1010110 in decimal notation.
3. (no points) Do you know of any particular historical significance to the numbers in 1 and 2? ["Get Smart" agents]
4. Let a = 49 and b = 11. Find q and r such that a=bq+r and r is in the range 0, 1, ..., 10.

Quiz 1 (Due 1/24. Practice, to test system.)