Prove that the sum of two odd numbers is always even
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Step-by-Step Solution
Step 1: Define Odd Numbers
To prove this statement, we first need to define what an odd number is. An odd number is any integer that cannot be divided exactly by 2. Mathematically, we can represent any odd number as 2k+1, where k is an integer. We will use this definition for our two odd numbers, m and n.
Step 2: Represent the Odd Numbers Algebraically
Since m and n are two different odd numbers, we can represent them using the definition of an odd number. We use different integer variables, k1 and k2, to ensure that m and n can be any two odd numbers, not necessarily the same one.
Step 3: Find the Sum of the Two Odd Numbers
Now that we have algebraic expressions for our two odd numbers, m and n, we can find their sum by adding these expressions together. This is the first step towards simplifying the sum.
Step 4: Simplify the Sum
We can simplify the expression for the sum by combining the constant terms. This brings us closer to identifying the form of an even number.
Step 5: Factor out 2
To show that the sum is an even number, we need to demonstrate that it can be written in the form 2×integer. We can achieve this by factoring out 2 from the simplified sum.
Step 6: Conclude the Proof
Since k1 and k2 are integers, their sum k1+k2 is also an integer. Adding 1 to this sum, K=k1+k2+1, also results in an integer. Thus, the sum m+n can be expressed as 2K, which is the definition of an even number. This completes the proof.