Math115a (Youtube)

  1. Lecture 03/04: Divisibility and Euclid's Algorithm
  2. Lecture 05: Primes
  3. Lecture 06: Multiplicative Functions
  4. Lecture 07: Binomial Coefficients
  5. Lecture 08: Applications of Binomial Coefficients
  6. Lecture 09: Congruences
  7. Lecture 10: Applications of Fermat's Theorem
  8. Lecture 11: Euler's Theorem

Math115a (Attempted Problems)

  1. 1.2: Problem 00: Show that \([a,b] = \frac{ab}{(a,b)}\) for any integers \(a,b\).
  2. 1.2: Problem 19: Any set of integers that relatively prime in pairs are relatively prime.
  3. 1.2: Problem 21: Prove that if an integer of the form \(6k + 5\), then it is necessarily of the form \(3k-1\), but not conversely.
  4. 1.2: Problem 24: Prove that if \(n\) is composite, it must have a prime factor \(p \leq \sqrt{n}\).
  5. 1.2: Problem 26: Let \(s\) and \(g > 0\) be given integers. Prove that integers \(x\) and \(y\) exist satisfying \(x + y = s\) and \((x, y) = g\) if and only if \(g \mid s\).

  6. 1.3: Problem 06: Show that every positive integer \(n\) has a unique expression of the form \(n = 2^rm\), \(r \geq 0\), \(m\) a positive odd integer.
  7. 1.3: Problem 14: Evaluate \(ab, p^4\) and \(a+b, p^4\) given that \((a,p^2) = p\) and \((b, p^3) = p^2\) where \(p\) is prime.
  8. 1.3: Problem 16: Find a positive integer \(n\) such that \(n/2\) is a square, \(n/3\) is a cube, and \(n/5\) is a fifth power.
  9. 1.3: Problem 18: Prove that \((a^2,b^2) = c^2\) if \((a,b) = c\).
  10. 1.3: Problem 19: Let \(a\) and \(b\) be positive integers such that \((a,b) = 1\) and \(ab\) is a perfect square. Prove that \(a\) and \(b\) are perfect squares. Prove that the result generalizes to \(k\)th powers.
  11. 1.3: Problem 20: Given \((a,b,c)[a,b,c] = abc\), prove that \((a,b) = (b,c) = (a,c) = 1\).

  12. 1.4: Problem 04: Suppose that \(S\) contains \(2n\) elements and that \(S\) is partitioned into \(n\) disjoint subsets each one containing exactly two elements of \(S\). Show that this can be done in precisely

  13. 2.1: Problem 02: Exhibit a complete residue system module 17 composed entirely of multiples of 3.
  14. 2.1: Problem 06: Prove that if \(p\) is a prime and \(a^2 \equiv b^2 (\bmod p)\), then \(p \mid (a + b)\) or \(p \mid (a - b)\)
  15. 2.1: Problem 08: Prove that any number that is a square must have one of the following for its units digit: \(0,1,4,5,6,9\)
  16. 2.1: Problem 09: Prove that 19 is not a divisor of \(4n^2 + 4\) for any integer \(n\).
  17. 2.1: Problem 09: Show that if \(p \equiv 3 (\bmod 4)\), then \(\frac{p-1}{2}! \equiv \pm (\bmod p)\)

Other Notes

  1. Multiplicative Order
  2. Reptend Prime
  3. Euler Totient's Function