| Math Mojo - Making Math Meaningful |
A Short Lesson about The Products of two Even or two Odd Integers
1).
Prove that the product of two even integers equal an even integer.
2).Prove that the product of two odd integers equal an odd integer.
1) You know that the product
of any two integers will end in the digit of the product of the digits
in the ones columns of the two integers.
For example: 34*28 must end
in 2, because 4*8 is 32, which ends in 2.
Therefore, our proof need only be
limited to all possible products of any two even digits.
That leaves us with only the following
possibilities to have to prove:
2*2, 2*4, 2*6, 2*8, 4*4, 4*6,
4*8, 6*6, 6*8, and 8*8.
By inspection, you see that all of
them end with an even digit. Therefore any even integer times any even integer
will always give you an even integer as the product.
2). Use the same kind of argument
for the products of two odd integers being odd.
The first part of the argument is almost exactly the same:
You know that the product of any two integers will end in the digit of the product
of the digits in the ones columns of the two integers.
Therefore, our proof need only be limited to all possible products of any two
odd digits.
That leaves us with only these possibities to have to prove:
1*1, 1*3, 1*5, 1*7, 1*9, 3*3, 3*5, 3*7, 3*9, 5*5, 5*7, 5*9, 7*7, 7*9, and 9*9.
By inspection, you see that all of them end with an odd digit. Therefore any
odd integer times any odd integer will always give you an odd integer as the
product.
Q.E.D.
For further thought:
What do you think the product of three even or three odd integers will be; odd or even? Can you prove it? How about four of each? Can you make up a general rule from your findings, and can you prove it?
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