1. Clements MAΘ October 30th, 2014
NUMBER THEORY
Clements MAΘ
October 30th, 2014

2. MODULAR CONGRUENCES
Let a,b be integers and let m be a positive integer. We say that a and b are congruent modulo m if m|a - b. This is denoted as a≡b (mod m). If m∤a - b, we write
a≢b (mod m).
x|y means x divides y, i.e. there is an integer you can multiply x by to get y.
4 divides 12, but 4 doesn’t divide 6.

3. EXAMPLES
5≡7 (mod 2)
4≡13 (mod 3)
6≡21 (mod 5)
-2≡10 (mod 12)

4. Mods can be added, subtracted, multiplied, and taken to powers:
PROPERTIES
Mods can be added, subtracted, multiplied, and taken to powers:
If ai ≡ bi (mod m) for 1 ≤ i ≤ n, then
a1 + a2 + a3 + … + an ≡ b1 + b2 + b3 + … + bn (mod m).
If a + b ≡ c (mod m), then a ≡ c - b (mod m).
If a ≡ b (mod m)
a + c ≡ b + c (mod m).
ac ≡ bc (mod m).
an ≡ bn (mod m) for all naturals n.
f(x) is a polynomial with integer coefficients, then f(a) ≡ f(b) (mod m).

5. PROPERTIES (cont.) Reflexive, Symmetric, Transitive a≡a (mod m)
b≡a (mod m) and a≡b (mod m)
a≡b and a≡c then a≡c (mod m)
Equivalence relation: m equivalence classes
mn+0 , mn+1 , mn+2 … mn + (m-1)
Assuming equivalence class of an integer a is [a]:
[a+b] = [[a] + [b]]
[a*b] = [[a] * [b]]

6. APPLICATIONS Given 3a – 3 = 12, prove a must be odd.
Prove there are no solutions to:
2a – 3 = 12
Prove there are no solutions to the system:
2a + 3b = 3
6a – 5b = 4

7. FERMAT’S LITTLE THEOREM
If p is a prime and a is an integer, then
ap-1 ≡ 1 (mod p).

8. PROOF 1

9. PROOF 1

10. PROOF 2

11. PROOF 2

12. Problems Find the last digit of Find its last two digits.
Can a square have last digits 85? E.g. could 485 or 5485 be squares?
What’s a fast way to check if a number is divisible by 9? Are these numbers multiples of 9?
62315
Is there a rule for 11 or 13?