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Math 2534 Test 1B Solutions
Problem 1: Using one or more of the following previously proved theorems below,
prove the following Theorem:
List of previously proved theorems:
Refer to needed theorems only. Do not try to reprove any of these theorems.
Two consecutive integers have opposite parity
The product of two even (odd) integers is even (odd).
The sum of any two odd integers or any two even integers is even.
Any prime number greater than 2 is an odd integer.
The sum of any two integers is an integer.
The sum of an even and odd integer is odd.
Solutions: (12pts):
4
2 is a prime number and ( 1) is an even number, then is even.If a b a b>+ +
Proof: If a is a prime number greater than 2, then a is odd. If a is odd then a
2
= (a)(a) is
also odd since the product of two odd numbers is always odd. We also conclude
that a
4
= a
2
a
2
is also odd since the product of two odd numbers is odd. We are given that
b+1 is even. Then b will be odd since...

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