Introductory Number Theory
MIDTERM EXAM
Solutions
Instructions.
•
Use your own paper, or the one I provide at the front. Write clearly. Keep different exercises separate; that
is, try to finish one exercise before starting another, and make it clear where one exercise ends and the other
begins. Do not try to cram as much as you can into a single page; there is no prize for that. Nor is there a
prize for writing as small as possible.
•
All exercises are worth 20 points. To get the 20 points, the exercise must have been done completely. Partial
credit will be given for partial work, but only if the partial work is a substantial part of what would be the
whole work.
The Test.
1. Find all primitive Pythagorean triples (
a, b, c
) with
a
+
b
+
c
= 70. For full credit, you must list all such triples
(assuming there are any; a possible answer is there are none), and (more importantly) prove that there are
no others.
Solution.
If (
a, b, c
) is a primitive Pythagorean triple, then
a
=
st
,
b
= (
s
2
−
t
2
)
/
2,
c
= (
s
2
+
t
2
)
/
2, where
s, t
are odd
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 Summer '11
 STAFF
 Number Theory, Natural number, Prime number, Euclidean algorithm, Pythagorean triple, prime power factorization

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