The product of two odd integers is always odd. See solution.
Practice makes perfect
Let's attempt to find a pattern by calculating the product of two odd integers.
(1)(3)&=4 && (3)(5)=15
(5)(7)&=35 &&(9)(11)=99
Examining the sums, we see that all of them are even. We can make a conjecture by inductive reasoning.
Conjecture: & The product of two odd
&integers is always odd.
To use deductive reasoning, we have to prove this is true for all integers. If we let a and b be arbitrary integers, then 2a and 2b will always be even, since all even integers are divisible by 2. To make these into odd integers, we add 1 to them. With this information we can write two arbitrary odd integers.
Odd Number1: 2a+1
Odd Number2: 2b+1
Let's add them and see what happens.
As we can see, the product of two arbitrary odd numbers can be rewritten as a sum of an even number and 1. As we already explained, this is by definition an odd number. This means our conjecture is true.
