Explanation:

To find the smallest number that fits the criteria, we can use the Chinese Remainder Theorem or simple trial and error to check each option.

- The conditions are: when divided by 19, the remainder is 13. When divided by 23, the remainder is 17.

- Option 1: 437

- 437 divided by 19 gives a remainder of 0.

- Not meeting the criteria for 19.

- Option 2: 419

- 419 divided by 19 gives a remainder of 1.

- Not meeting the criteria for 19.

- Option 3: 431

- 431 divided by 19 gives a remainder of 13.

- 431 divided by 23 gives a remainder of 17.

- Meets both conditions.

- Option 4: 429

- 429 divided by 19 gives a remainder of 11.

- Not meeting the criteria for 19.

The correct answer is Option 3: 431.

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