Explanation:

Let's analyze each option:

- The number leaves a remainder of 19 when divided by 34 or 51. So, the number can be written as:

Number = 34k + 19 = 51m + 19, for integers k and m.

- This means the number minus 19 must be divisible by 34 and by 51.

Let's check each option:

1. Option 1: 887

- 887 - 19 = 868

- 868 ÷ 34 = 25.529… Not an integer

- 868 ÷ 51 = 17.019… Not an integer

2. Option 2: 821

- 821 - 19 = 802

- 802 ÷ 34 ˜ 23.59 Not an integer

- 802 ÷ 51 ˜ 15.73 Not an integer

3. Option 3: 835

- 835 - 19 = 816

- 816 ÷ 34 = 24 Exactly divisible

- 816 ÷ 51 = 16 Exactly divisible

- 835 is the number which fits the condition.

4. Option 4: 869

- 869 - 19 = 850

- 850 ÷ 34 ˜ 25 Not an integer

- 850 ÷ 51 ˜ 16.67 Not an integer

Correct Answer:

Option 3: 835