To show that any positive odd integer can be represented in the form of 6q + 1, 6q + 3, or 6q + 5, let’s first understand what odd integers are and how they relate to modulo operations.
1. **Definition of Odd Integers:** An integer is considered odd if it cannot be evenly divided by 2. Thus, the general formula for odd integers can be expressed as:
n = 2k + 1
(where k is any integer)
2. **Using Modular Arithmetic:** When dealing with the expression modulo 6, we can categorize odd numbers. Since we’re considering integers in relation to 6, let’s express any integer in the form of:
n = 6q + r
(where r is the remainder when n is divided by 6, and r can take values from 0 to 5)
3. **Examining Odd Remainders:** We need to find which of these remainder cases produce odd numbers:
- When r = 0, then n is even (6q + 0).
- When r = 1, then n is odd (6q + 1).
- When r = 2, then n is even (6q + 2).
- When r = 3, then n is odd (6q + 3).
- When r = 4, then n is even (6q + 4).
- When r = 5, then n is odd (6q + 5).
From this breakdown, we can observe that there are three cases where the integer is odd:
- 6q + 1
- 6q + 3
- 6q + 5
4. **Conclusion:** Therefore, any positive odd integer can indeed be expressed in one of these forms. For example:
- The number 1 can be represented as 6*0 + 1.
- The number 3 can be represented as 6*0 + 3.
- The number 5 can be represented as 6*0 + 5.
- The number 7 can be represented as 6*1 + 1.
- The number 9 can be represented as 6*1 + 3.
- The number 11 can be represented as 6*1 + 5.
In summary, every positive odd integer must fit one of the forms 6q + 1, 6q + 3, or 6q + 5 based on modular arithmetic, confirming the statement.