If $$ x $$ is an odd integer, which of the following statements is correct?

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Accepted Answer

If $$ x $$ is an odd integer, it can be expressed as $$ x = 2k + 1 $$, where $$ k $$ is an integer.

1. **Step 1: Evaluate $$ x^2 $$:**  
   Substituting $$ x = 2k + 1 $$,  
   $$
   x^2 = (2k + 1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1
   $$  
   Since $$ x^2 = 2m + 1 $$ (where $$ m = 2k^2 + 2k $$), $$ x^2 $$ is odd.

2. **Step 2: Evaluate $$ 5x^2 $$:**  
   Multiplying by 5,  
   $$
   5x^2 = 5(2m + 1) = 10m + 5
   $$  
   This is also odd because it can be expressed as $$ 2n + 1 $$, where $$ n = 5m + 2 $$.

3. **Step 3: Evaluate $$ 5x^2 + 2 $$:**  
   Adding 2 to $$ 5x^2 $$,  
   $$
   5x^2 + 2 = (10m + 5) + 2 = 10m + 7
   $$  
   This result is odd since it can still be expressed in the form $$ 2p + 1 $$, where $$ p = 5m + 3 $$.

4. **Step 4: Evaluate $$ 5x^2 + 3 $$:**  
   Adding another odd number (3),  
   $$
   5x^2 + 3 = (10m + 5) + 3 = 10m + 8
   $$  
   This result is even since it can be written as $$ 2q $$, where $$ q = m + 4 $$.

Thus, the correct statement is **Option (2): $$ 5x^2 + 2 $$ is odd**.

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