Final answer:
The product of any two odd numbers x (2k+1) and y (2m+1) is odd, as shown by the algebraic proof that multiplies both and rearranges to show the product is of the form 2n+1, which defines an odd number.
Step-by-step explanation:
To prove that the product of two odd numbers is odd, let's define odd numbers more formally. An odd number can be expressed as 2k+1 where k is any integer. Therefore, let's denote x and y as two odd numbers:
x = 2k + 1
y = 2m + 1
where k and m are integers.
When we multiply these two odd numbers together:
xy = (2k + 1)(2m + 1)
= 2k(2m) + 2k + 2m + 1
= 4km + 2k + 2m + 1
= 2(2km + k + m) + 1
Notice that 2km + k + m is an integer since it is a sum of integers. Therefore, we can denote this integer as n for simplicity:
xy = 2n + 1
Since 2n is even, 2n + 1 is an odd number. This is because adding 1 to an even number always results in an odd number. Hence, the product of x and y, both odd numbers, is indeed odd.