Algebraic Manipulation Worksheets With Answers - Serge Ballif

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Serge Ballif
MATH 567 Homework 1
Tuesday, September 2, 2008
Easier Problems
1. Given a b and c d, prove that ac bd.
The expressions a
b and c
d mean that there exist x and y such that ax
b and
cy
d. Therefore, (ac)(xy)
bd and ac bd.
2
2. Prove that if n is odd, then n
1 is divisible by 8.
2
2
Since n is odd, n is of the form n
2m
1. Hence n
1
4m
4m
4m(m
1).
Either m or m 1 must be an even number (of the form 2k for some k). Therefore, 8 is
2
a factor of n
1.
3. Find the greatest common divisor g of the numbers 1819 and 3537, and then find integers x and y to satisfy
1819x
3587y
g
We use the Euclidean algorithm to find the gcd.
3587
1819(1)
1768
1819
1768(1)
51
1768
51(34)
34
51
34(1)
17
34
17(2)
0
Thus (3587 1819)
17. Now we back-substitute starting from the 4th line.
17
51
34
51
(1768
51(34))
51(35)
1768
(1819
1768)(35)
1768
1819(35)
1768(36)
1819(35)
(3587
1819)(36)
1819(71)
3587(36)
Hence we can choose x
71 and y
36.
2
4. Let g
0 and b be given integers. Prove that the equations (x y)
g and xy
b can be solved if and only if g
b.
( ) Suppose that (x y)
g and xy
b can be solved. Then g x and g y implies that
2
2
x
ga and y
gb for some integers a and b. Then b
xy
(ab)g
, so g
b.
2
( ) Suppose that g
and b
are numbers such that g
b. Then for some integer
2
a we have the equation ag
b. The equations (x y)
g and xy
b are satisfied by
the values x
g and y
ag.
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