Integer Worksheet
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1Math 2534:
In Class Problem Sheets: Wednesday Feb 9
Prove or give counter examples for the following:
2
1)
Theorem: Suppose n is an integer, If n is even then n
is even.
→
2)
Theorem: Given a, b, c are integers, If a bc
a b
+
→
3)
Theorem: Given a, b, c are integers, If
a b and a b c
,
(
)
a c
4)
Theorem: The product of two rational numbers is a rational number.
2
2
5)
Theorem:
Given an integer n, If n
is even then n
is divisible by 4.
2
6)
Theorem:
If n is a natural number then n
+ n is even.
7)
Theorem: Given n is an integer, n is even if and only if 7n + 4 is even.
3
8)
Theorem: Suppose n is an integer, If n
+ 5 is odd then n is even.
3
9)
Theorem: If n is a natural number then n
– n is even.
+
≠
3
3
3
10
Theorem: If a, b, c are prime numbers greater than 2, Then
a
b
c
−
−
+
− =
3
2
11)
Theorem: If x is a real number, x = 1 if and only if
x
3
x
4
x
2 0
12)
Theorem: 1 = 2
2
2
2
2
Proof: Consider two equal integers, a = b. Then a
= ab and a
-b
= ab – b
.
This will now give us (a – b)(a + b) = b(a – b) and dividing both sides by
(a – b) will give a + b = b. Since a = b , we have that 2b = b and
therefore 2 = 1. QED
Is this proof valid? Explain your reasoning process.
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