Consider the following statements: The numbers 24984, 26784 and 28584 are (1) divisible by 3 (2) divisible by 4 (3) divisible by 9 Which of these are correct?

### Concept & Strategy This question requires verifying three separate divisibility rules across three different numbers. * Rule for 4: The last two digits form a number divisible by 4. * Rule for 3: The sum of digits is divisible by 3. * Rule for 9: The sum of digits is divisible by 9. ### Step-by-Step Solution Let us evaluate the statements systematically. * **Test Statement (2): Divisible by 4** Look at the last two digits of all three numbers: 24984, 26784, 28584. They all end in 84. Since 84 is a multiple of 4 ($84 / 4 = 21$), all three numbers are perfectly divisible by 4. Statement (2) is correct. * **Test Statements (1) and (3): Divisible by 3 and 9** We need to find the sum of the digits for each number. * For 24984: $2 + 4 + 9 + 8 + 4 = 27$ * For 26784: $2 + 6 + 7 + 8 + 4 = 27$ * For 28584: $2 + 8 + 5 + 8 + 4 = 27$ The sum for all three numbers is exactly 27. Since 27 is divisible by 3, statement (1) is correct. Since 27 is divisible by 9, statement (3) is correct. Because statements (1), (2), and (3) are all valid, the correct option encompasses all of them. ### Exam Strategy & Shortcut Notice the pattern in the given numbers: 24984, 26784, 28584. The only digit changing is the thousands digit, which increases by 2 each time (4 to 6 to 8), while the hundreds digit decreases by 2 each time (9 to 7 to 5). This means the sum of the digits remains absolutely constant. You only need to calculate the sum for the very first number (27) to know the sum for all three, saving you from doing three separate additions. ### Common Pitfall Students often check only one number (e.g., just 24984) and assume the rest follow suit without verifying if the properties hold true for all items in the list. While it worked out here due to the hidden arithmetic pattern, blindly assuming can lead to trap answers. ### Final Answer Therefore, the correct answer is **1, 2 and 3**.