2
You can use
MAX(a,b) + MIN(a,b) = a+b
applied appropriately to the
factorizations of x and y to prove
the above fact…
Fact:
GCD(x,y)
×
LCM(x,y) = x
×
y
(a mod n)
means the
remainder
when
a is divided by n.
a mod n = r
a = dn + r for some integer d
Definition: Modular equivalence
a
b [mod n]
(a mod n) = (b mod n)
n | (a-b)
a = q n + b for some q
Written as a
n
b, and
spoken
“a and b are
equivalent
or
congruent
modulo n”
31
81 [mod 2]
31
2
81
31
80 [mod 7]
31
7
80
n
is an equivalence relation
In other words, it is
Reflexive:
a
n
a
Symmetric:
(a
n
b)
(b
n
a)
Transitive:
(a
n
b and b
n
c)
(a
n
c)
n
induces a natural partition of the
integers into n “residue” classes.
(“residue” = what’s left over = “remainder”)
Define
residue class
[k]
=
the set of all integers
that
are congruent to
k modulo n
.
Residue Classes Mod 3:
[0]
= { …,
-6, -3, 0, 3, 6, .
.}
[1]
= { …,
-5, -2, 1, 4, 7, .
.}
[2]
= { …,
-4, -1, 2, 5, 8, .
.}
[-
6] = { …,
-6, -3, 0, 3, 6, .
.}
[7]
= { …,
-5, -2, 1, 4, 7, .
.}
[-
1] = { …,
-4, -1, 2, 5, 8, .
.}
= [0]
= [1]
= [2]