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
  9. Lecture 12: Wilson's Theorem
  10. Lecture 13: Chinese Remainder Theorem
  11. Lecture 14: Euler Totient Function
  12. Lecture 15: Numerical Calculations [TODO]

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 12: Prove that 19 is not a divisor of \(4n^2 + 4\) for any integer \(n\).
  17. 2.1: Problem 17: Show that \(61! + 1 \equiv 63! + 1 \equiv 0 \pmod{71}\).
  18. 2.1: Problem 18: Show that if \(p \equiv 3 \pmod{4}\), then \(\left(\frac{p-1}{2}\right)! \equiv \pm 1 \pmod{p}\)
  19. 2.1: Problem 19: Prove that \(n^6 - 1\) is divisible by \(7\) if \((n,7) = 1\).
  20. 2.1: Problem 20: Prove that \(n^7 - n\) is divisible by \(42\) for any integer \(n\).
  21. 2.1: Problem 21: Prove that \(n^{12} - 1\) is divisible by \(7\) if \((n,7) = 1\).
  22. 2.1: Problem 23: Prove that \(n^{13} - n\) is divisible by \(2,3,5,7\) and \(13\) for any integer \(n\).
  23. 2.1: Problem 21: Prove that \(n^{12} - a^{12}\) is divisible by \(13\) if \(n\) and \(a\) are prime to \(13\).
  24. [TODO] 2.1: Problem 21: Prove that \(\frac{1}{5}n^{5} + \frac{1}{3}n^{3} + \frac{7}{15}n\) is an integer for every integer \(n\).

Other Notes

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