CSE
1400
Applied Discrete Mathematics
Modular Numbers
Department of Computer Sciences
College of Engineering
Florida Tech
Fall
2011
Conversion Between Units
1
Modular Numbers
3
Division mod n
5
Linear Congruence Equations
6
Euclidean Algorithm for the Greatest Common Divisor
7
Relatively Prime Numbers
10
Extended Euclidean Algorithm
10
Solving Linear Congruence Equations
14
Problems on Modular Numbers
17
Abstract
Conversion Between Units
P
retend you
’
d like to covert one foot
(12
inches
)
into its
equivalent length in meters
.
You know one foot goes into
The conversion between units is
1 meter
≈
3.2808 feet
1 foot
≈
0.3048 meters
The
Euclidean algorithm
compute these
approximations.
one meter about 3 times with a remainder of about 3 inches.
The
1 foot
1 meter
empirical approach to convert feet to meters is to lay the footlong
ruler along a meter stick as many times as it will go; in this case 3
times.
One meter is approximately three feet.
1 meter
=
3 feet
+
r
feet
where
r
is some fractional part of one
foot. Three, the number of time a foot
goes into a meter is called the quotient,
and
r
is called the remainder or residue.
1 foot
1 meter
There will be a tiny piece
left over, not covered, on the meter stick.
Call the length of this tiny piece
r
and note that
r
is some fraction of
a foot.
To compute
r
, lay it along the footlong ruler and measure that
about three
r
’s almost covers a foot, so that
r
≈
0.3333
· · ·
feet and 1
meter is approximately 3.3333
· · ·
feet.
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cse 1400 applied discrete mathematics
modular numbers
2
1 foot
But there is still a tiny piece
left over, not covered, on the footlong
ruler. Call the length of this tiny piece
r
1
, and let’s also rename
r
, the
first fraction of a foot, by calling it
r
0
from now on.
To compute the length
r
1
, lay it along the fractional foot
r
0
=
≈
1
3
and measure that two of these smaller remainder
r
1
=
almost cover
r
0
=
, but by a little too much.
We can continue this process until:
• Some multiple of a tiny remainder exactly covers the previous one.
• The approximation computed is accurate enough for its intended
purpose.
Let’s stop at this step and note what we’ve discovered.
1 meter
= (
3
+
r
0
)
feet
1 foot
= (
3
r
0
+
r
1
)
feet
r
0
feet
= (
2
r
1

r
2
)
feet
We make a truncation error by ignoring the value of
r
2
, effectively
setting its value to 0, but we gain a conversion factors to change
meters into feet and vice versa. That is, we have
r
1
feet
≈
r
0
2
feet
1 foot
≈
(
3
r
0
+
r
0
2
) =
7
r
0
2
feet
r
0
feet
≈
2/7 feet
1 meter
≈
(
3
+
2
7
)
feet
Or, 1 meter is about 23/7
=
3.28571428571 feet. Let us declare that
3
+
2/7
=
23/7
≈
3.2857 feet is “good enough for our uses” as an
approximation to 1 meter.
Recall from
above
, 1 meter is approxi
mately 3.2808 feet.
In practice we would like to have a ruler 1/7 of a footlong so we
subdivide 1 foot evenly into 7 parts and 1 meter (almost) evenly into
23 parts. Here’s how we can
construct
a rule oneseventh of a
footlong. Lay out three meters and you’ll discover you’ve covered
3
(
3
+
2/7
) =
9
+
6/7 feet. Now lay out ten footlong rulers to 10 feet.
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 Fall '11
 Shoaff
 Math, Division, Equations, Congruence, Prime Numbers, Greatest common divisor, Euclidean algorithm, Linear Congruence Equations

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