1. Use Euclid’s Algorithm to find the gcd of each pair
a) 24,138 b) 159 , 414 c) 272 and 1479 d) 4144 and 7696
2. Use the prime factorization of the numbers in Problem 1 to find their gcd’s.
(feel free to use software for the prime factorization)
3. a) If a|b and a|c prove that a|(b+c).
b) prove gcd(ta,tb) = t gcd(a,b)
4. A student claims that (2n+1, 2n2 + 2n, 2n2 + 2n + 1) give Pythagorean Triples
a) verify it does generate Pythagorean Triples (i.e satisfy the Pythagorean Theorem)
b) can you find one triple that it does not generate? (this would then prove that it only produces a proper subset)
5. Argue that the sum of the squares of two odd numbers must always be even.
6. What is the sum of
100+103+106+ . . . + 1399 ?
7. For a general arithmetic series with n+1 terms a + (a+d) + (a+2d) + . . . (a+nd) Can you work out a formula for the sum?
n
2 n(n +1)(2n +1)
8. Can you prove the arithmetic series formula ∑i=1i = 6 by induction?
9. What are the possible values of gcd(n,n+2) where n is any possible positive integer ?
10. a) Use Geometric series to sum the first n powers of 2 1 + 2 + 22 + . . +2n-1
a) What is the binary form of 27 -1 ? 29-1 ? 2n -1 ? (by binary is meant base 2)
11. Consider numbers of the form 2pq - 1 where p and q are integers greater than 1. See if you can factor this. (hint: use Geometric Series )
12. Write Maple or Matlab code to generate a list of PTs as discussed in Problem #4. Do a screen shot and include it in your work turned in. Which ones are PPTs ?
