Ma 450: Mathematics for Multimedia
Solution:
to Homework Assignment 1
Prof. Wickerhauser
Due Friday, February 5th, 2010
1. Suppose
a
divides 2
b
and 2
b
divides
c
. Must
a
divide
c
?
Solution:
Yes. Rename
B
= 2
b
and apply the transitive property of divisibility to
a, B, c
.
2. Write a computer program that finds the greatest common divisor of three positive integers
a, b, c
,
assuming that the greatest common divisor function
gcd(x,y)
for any two positive integers
x,y
has
already been implemented.
Solution:
Apply
gcd(x,y)
twice:
Standard C Function: Greatest Common Divisor of Three Integers
int gcd3 ( int a, int b, int c ) {
a = gcd(a,b);
return gcd(a,c);
}
3. Suppose that
a
+
b
and
a

b
are relatively prime. Prove that
a
and
b
must be relatively prime.
Solution:
Since gcd(
a
+
b, a

b
) = 1, by Theorem 1.2 there are integers
x, y
such that 1 =
x
(
a
+
b
) +
y
(
a

b
) = (
x
+
y
)
a
+ (
x

y
)
b
.
But then 1 is the smallest positive element of the set
{
ma
+
nb
:
m, n
∈
Z
}
, so gcd(
a, b
) = 1 as in the proof of Theorem 1.2.
4. Find the greatest common divisor of the two numbers 123 456 789 and 12 345 678 901 234 567 890.
Solution:
The larger of these numbers is evidently a multiple of the smaller, so the greatest common
divisor is just the smaller number, 123 756 789, by the fourth useful fact on page 4.
5. Is there an integer
x
such that 3702
x

1 is divisible by 85?
Solution:
Yes. Such an
x
exists if there is a solution to the equation 3702
x

1
≡
0
(mod 85), or
3702
x
≡
1
(mod 85). But gcd(3702
,
85) = 1, so by Lemma 1.6 such a quasiinverse exists.
We may use the extended Euclid algorithm to find the solution
x
= 38: (3702)(38)

1 = 140675 =
(85)(1655).
6. Suppose that

2
30
< x <
0 and

2
30
< y <
0 for integers
x, y
. Is it possible that the operation
x
+
y
causes integer underflow on a 32bit twos complement computer?
Solution:
No.
Integer underflow occurs on such a computer if and only if the sum of two rep
resentable negative numbers
x, y
gives the 32bit twos complement representation of a nonnegative
1
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 Spring '10
 Wickerhauser
 Math, Prime number, Rational number, Greatest common divisor

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