Using algebraic expressions, we can prove properties that always hold.
Representing Even and Odd Numbers
- Even number: 2n (n is an integer)
- Odd number: 2n + 1
- Consecutive integers: n、n+1、n+2、…
- Consecutive even numbers: 2n、2n+2、2n+4、…
- Consecutive odd numbers: 2n+1、2n+3、2n+5、…
Example ①
Explain why the sum of an even number and an odd number is odd.
Answer:Let the even number be 2m and the odd number be 2n+1. Sum: 2m+2n+1 ＝ 2(m+n)+1. Since m+n is an integer, the sum is odd.
Explain why the sum of an even number and an odd number is odd.
Answer:Let the even number be 2m and the odd number be 2n+1. Sum: 2m+2n+1 ＝ 2(m+n)+1. Since m+n is an integer, the sum is odd.
Proving Multiples
Example ②
Explain why the sum of two consecutive odd numbers is a multiple of 4.
Answer:Let the numbers be 2n+1 and 2n+3. Sum: 4n+4 ＝ 4(n+1). Since n+1 is an integer, the sum is a multiple of 4.
Explain why the sum of two consecutive odd numbers is a multiple of 4.
Answer:Let the numbers be 2n+1 and 2n+3. Sum: 4n+4 ＝ 4(n+1). Since n+1 is an integer, the sum is a multiple of 4.
Representing a Two-Digit Number
Let a be the tens digit and b the ones digit → the two-digit number is 10a + b. Reversing the digits gives 10b + a.
Example ③
Explain why the difference between a two-digit natural number and the number obtained by reversing its digits is a multiple of 9.
Answer:(10a+b)−(10b+a) ＝ 9a−9b ＝ 9(a−b)。Since a−b is an integer, the difference is a multiple of 9.
Explain why the difference between a two-digit natural number and the number obtained by reversing its digits is a multiple of 9.
Answer:(10a+b)−(10b+a) ＝ 9a−9b ＝ 9(a−b)。Since a−b is an integer, the difference is a multiple of 9.
Key Points
- Even number: 2n、odd number: 2n+1
- To show that a number is a multiple of ○, write it as ○×(integer)
- A two-digit number is 10a+b
- Finally, add the statement “○ is an integer”
Practice Problems
- Explain why the sum of two odd numbers is even.
- Explain why the sum of three consecutive integers is a multiple of 3.
- Explain why the sum of a two-digit number and the number obtained by reversing its digits is a multiple of 11.
Answers and Explanations
- Answer:Let the numbers be 2m+1 and 2n+1. Sum: 2(m+n+1). Therefore, it is even.
Explanation: It has the form 2×(integer)。 - Answer:Let the numbers be n、n+1、n+2. Sum: 3n+3 ＝ 3(n+1). It is a multiple of 3.
Explanation: Factor out 3。 - Answer:(10a+b)+(10b+a) ＝ 11(a+b)。It is a multiple of 11.
Explanation: Factor out 11。