Modular Arithmetic
Definition:
If
a, b
∈
Z
, and
m
∈
Z
+
, we say that
a
is congruent to
b
modulo
m
, and write
a
≡
b
(mod
m
) if
a
and
b
leave the same remainder on division by
m
.
Examples
. 10
≡
24 (mod 7),

15
≡
12 (mod 9), and 442
≡
2 (mod 10).
You can think of the integers (mod
m
) as the hours on a circular clock with
m
hours,
0
,
1
, . . . , m

1. For positive numbers you move around the circle clockwise, and for negative
numbers you move counterclockwise. The important part is that the number of times you
go around the circle and return to zero makes no difference to where you end up. What
matters is the number of places you move when it is no longer possible to make it around
the circle any more, and this number is one of 0
,
1
,
2
, . . . , m

1. These are the possible
remainders on division by
m
.
The universe of integers (mod
m
) really only consists of the numbers 0
,
1
,
2
, . . . , m

1;
modulo
m
, any other integer is just one of these with another name.
In many programming languages there is a function
mod
. If
m
= 0 is an integer, then
a
(mod
m
) is the unique number among 0
,
1
,
2
, . . . , m

1 to which
a
is congruent modulo
m
.
It is the remainder (as in the division algorithm  that’s why its unique) when
a
is
divided by
m
.
Questions
.
Prove that a nonnegative integer is congruent to its last digit (its ones
digit)
(mod 10). What is a nonnegative integer congruent to
(mod 100)?
(mod 1000)?
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 Spring '11
 stephenlang
 Calculus, Division, Remainder, 1K, two digits, m. Questions

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