a We will first examine the conjecture that Ayana makes.
The sum of two odd integers is an even integer.
Let's list some information that supports her conjecture.
Examples
1+3=4 ⇒ 4 ÷ 2 =2
3+5=8 ⇒ 8 ÷ 2 =4
5+7=12 ⇒ 12 ÷ 2 =6
7+9=16 ⇒ 16 ÷ 2 =8
9+11=20 ⇒ 20 ÷ 2 =10
We gave five examples that support the conjecture. However, finding examples that support the conjecture does not prove that conjecture is true. We must show that the conjecture is true for all cases, not just a few.
b We have been told that two odd integers can be represented by the expressions 2n-1 and 2m-1, where n and m are both integers.
Odd number: 2 n-1, where n∈ Z
Odd number: 2 m-1, where m∈ Z
Let's give some examples that support the statement.
Examples
7=2( 4)-1
21=2( 11)-1
53=2( 27)-1
111=2( 56)-1
199=2( 100)-1
c If a number is even, we can say that it is a multiple of 2. When we look at Part A, we see that each example is divisible by 2. Moreover, in Part B, we can add the given expressions and show that the sum is divisible by 2.
d Let two odd integers be represented by 2n-1 and 2m-1. Let's add these two integers and check whether the sum has 2 as a factor.
