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86 The Nuts and Bolts of Proof, Third Edition
Therefore, the Hnes have in common the point with coordinates (1/5, 7/5).
This point is unique because its coordinates represent the only solution of
the system formed by the equations of the two lines. •
EXAMPLE 4. If a number is even, then its second power is divisible by 4
and its sixth power is divisible by 64.
Proof:
Hypothesis:
A. The number n is even.
{Implicit hypothesis: All the properties and operations of integer numbers
can be used.)
Conclusion:
B. The number n^ is divisible by 4.
C. The number n^ is divisible by 64.
Parti. If A, then B.
By hypothesis the number n is even. Therefore, n = 2t for some integer
number t. This implies that:
n^ = 4t\
As the number t^ is an integer, it is true that n^ is divisible by 4.
Part 2. If A, then C.
This implication can be proved in two ways.
First way: By hypothesis the number n is even. Therefore, n = 2t for
some integer number t. This implies that:
n^ = 64t\
As the number t^ is an integer, it is true that n^ is divisible by 64.
Second way: We can use the result estabhshed in Part 1, as that part of
the proof is indeed complete. When n is even, then n^ = 4k for some integer
number k. Then we have:
n^ = (n^ = (4ky = 64k\
Because k^ is an integer, it is true that n^ is divisible by 64.
•
2. If A, then B or C.
In this case, we need to show that given A, then either B or C is true
(not necessarily both). This means that we need to prove that at least one