Question

Show that any positive odd integer is of the form \( 4q + 1 \) or \( 4q + 3 \), where \( q \) is some integer.

Hint:

Any odd number, when divided by 4, will leave a remainder of 1 or 3.

Answer 1

Let \( n \) be any positive odd integer. Since \( n \) is odd, it can be expressed as: \[ n = 2k + 1, \quad \text{where} \quad k \in \mathbb{Z}. \] Now, we will divide \( n \) by 4. When we divide any integer by 4, the remainder can either be 0, 1, 2, or 3. Therefore, we can express \( n \) in one of the following forms: 1. If the remainder is 1 when divided by 4, then: \[ n = 4q + 1, \quad \text{where} \quad q \in \mathbb{Z}. \] 2. If the remainder is 3 when divided by 4, then: \[ n = 4q + 3, \quad \text{where} \quad q \in \mathbb{Z}. \] Thus, any positive odd integer can be written as \( 4q + 1 \) or \( 4q + 3 \), where \( q \) is some integer.
Conclusion:
Any positive odd integer is of the form \( 4q + 1 \) or \( 4q + 3 \), where \( q \) is some integer.