3, 5, 35, 10, 12, 35, 17, … choose which pair of numbers comes next?

The correct answer is $\boxed{\text{C. 19, 35}}$.

The sequence is generated by the following rule:

• Start with the numbers $3$ and $5$.
• Add $2$ to each number to get the next number in the sequence.
• Repeat step 2 until you reach the number $35$.
• Then, start over with the number $35$ and add $2$ to each number to get the next number in the sequence.

Therefore, the next two numbers in the sequence are $19$ and $35$.

Option A is incorrect because $22$ is not the next number in the sequence.

Option B is incorrect because $35$ is not the next number in the sequence.

Option C is correct because $19$ is the next number in the sequence.

Option D is incorrect because $19$ is not the next number in the sequence.

Option E is incorrect because $22$ is not the next number in the sequence.