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321 Section, 2-7

321 Section, 2-7. Natalie Linnell. Prove or disprove that if a, b, c are positive integers and a|bc then a|b or a|c. How many zeros are there at the end of 100!. Prove that if n is an odd positive integer, then n 2  1 (mod 8). Use Fermat’s Little Theorem to compute 3 302 mod 5.

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321 Section, 2-7

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  1. 321 Section, 2-7 Natalie Linnell

  2. Prove or disprove that if a, b, c are positive integers and a|bc then a|b or a|c

  3. How many zeros are there at the end of 100!

  4. Prove that if n is an odd positive integer, then n2 1 (mod 8)

  5. Use Fermat’s Little Theorem to compute 3302 mod 5

  6. Prove that if p is prime, and x2 = 1 (mod p) then x  1 (mod p) or x  (p-1) (mod p)

  7. Prove that if m and n are both perfect squares, then nm is a perfect square • What kind of proof did you do?

  8. Prove that if 3n+2 is odd, then n is odd • What kind of proof did you do?

  9. Show that the following is a tautology using a truth table • ((r→ (p v q)) → (p →(r →q))

  10. Let D(x,y) mean “team x defeated team y” and P(x,y) mean “team x has played team y” • Every team has lost at least one game. • There is a team that has beaten every team it has played.

  11. True or false: a  b (mod m), and b  c (mod m) implies that a2 bc (mod m)

  12. Is the argument correct? Why? • Every computer science major takes discrete mathematics. Natasha is taking discrete mathematics. Therefore, Natasha is a computer science major. • Everyone who eats granola every day is healthy. Linda is not healthy. Therefore, Linda does not eat granola every day.

  13. Show that ((p  q)  p)  qis a tautology using logical equivalences

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