No, the sum of any two odd numbers is not divisible by 4. It is divisible by 2, but not necessarily by 4. However, the sum of two consecutive odd numbers is divisible by 4.
Explanation
Let's represent odd numbers algebraically. Any odd number can be expressed as 2n + 1, where n is an integer.
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General Case (Any Two Odd Numbers):
Let's take two different odd numbers:
- Odd number 1: 2a + 1
- Odd number 2: 2b + 1 (where 'a' and 'b' are integers)
Their sum is: (2a + 1) + (2b + 1) = 2a + 2b + 2 = 2(a + b + 1). This sum is always divisible by 2, but not necessarily by 4. For example:
- 3 + 5 = 8 (divisible by 4)
- 3 + 7 = 10 (not divisible by 4)
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Specific Case (Two Consecutive Odd Numbers):
Consecutive odd numbers differ by 2. So, if the first odd number is 2n + 1, the next consecutive odd number is (2n + 1) + 2 = 2n + 3.
Their sum is: (2n + 1) + (2n + 3) = 4n + 4 = 4(n + 1). This sum is always divisible by 4, because 4 is a factor.
Examples:
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Not Divisible by 4 (Non-consecutive):
- 1 + 3 = 4 (divisible by 4 - happens to work, but not always)
- 1 + 5 = 6 (not divisible by 4)
- 3 + 7 = 10 (not divisible by 4)
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Divisible by 4 (Consecutive):
- 1 + 3 = 4 (divisible by 4)
- 3 + 5 = 8 (divisible by 4)
- 5 + 7 = 12 (divisible by 4)
- 7 + 9 = 16 (divisible by 4)
Conclusion
The sum of any two odd numbers is always divisible by 2, but it is only divisible by 4 if the two odd numbers are consecutive.
