Using the divisibility test, determine which of the following numbers are divisible by 2, 3, 4, 5, 6, 8, 9, 10, 11. Tabulate the data.

990; 1586; 275; 6686; 639210; 429714; 2856; 3060; 406839

To determine the divisibility of the given numbers by 2, 3, 4, 5, 6, 8, 9, 10, and 11, we can use specific rules for each divisor. Let’s break down these rules and apply them to the numbers you provided: 990, 1586, 275, 6686, 639210, 429714, 2856, 3060, and 406839. After that, we’ll summarize the results in a table for clarity.

Divisibility Rules Overview

• Divisible by 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8).
• Divisible by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
• Divisible by 4: A number is divisible by 4 if the last two digits form a number that is divisible by 4.
• Divisible by 5: A number is divisible by 5 if its last digit is 0 or 5.
• Divisible by 6: A number is divisible by 6 if it is divisible by both 2 and 3.
• Divisible by 8: A number is divisible by 8 if the last three digits form a number that is divisible by 8.
• Divisible by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
• Divisible by 10: A number is divisible by 10 if its last digit is 0.
• Divisible by 11: A number is divisible by 11 if the difference between the sum of its digits in odd positions and the sum of its digits in even positions is divisible by 11.

Applying the Rules

Now, let's apply these rules to each number:

1. Number: 990

• 2: Yes (last digit is 0)
• 3: Yes (9 + 9 + 0 = 18, which is divisible by 3)
• 4: Yes (last two digits 90, not divisible by 4)
• 5: Yes (last digit is 0)
• 6: Yes (divisible by 2 and 3)
• 8: No (last three digits 990, not divisible by 8)
• 9: Yes (sum of digits 18, divisible by 9)
• 10: Yes (last digit is 0)
• 11: No (9 - 9 + 0 = 0, not divisible by 11)

2. Number: 1586

• 2: Yes (last digit is 6)
• 3: No (1 + 5 + 8 + 6 = 20, not divisible by 3)
• 4: Yes (last two digits 86, not divisible by 4)
• 5: No (last digit is 6)
• 6: No (not divisible by 3)
• 8: No (last three digits 586, not divisible by 8)
• 9: No (sum of digits 20, not divisible by 9)
• 10: No (last digit is 6)
• 11: Yes (1 - 5 + 8 - 6 = -2, not divisible by 11)

3. Number: 275

• 2: No (last digit is 5)
• 3: No (2 + 7 + 5 = 14, not divisible by 3)
• 4: No (last two digits 75, not divisible by 4)
• 5: Yes (last digit is 5)
• 6: No (not divisible by 2)
• 8: No (last three digits 275, not divisible by 8)
• 9: No (sum of digits 14, not divisible by 9)
• 10: No (last digit is 5)
• 11: No (2 - 7 + 5 = 0, not divisible by 11)

4. Number: 6686

• 2: Yes (last digit is 6)
• 3: No (6 + 6 + 8 + 6 = 26, not divisible by 3)
• 4: Yes (last two digits 86, not divisible by 4)
• 5: No (last digit is 6)
• 6: No (not divisible by 3)
• 8: Yes (last three digits 686, divisible by 8)
• 9: No (sum of digits 26, not divisible by 9)
• 10: No (last digit is 6)
• 11: Yes (6 - 6 + 8 - 6 = 2, not divisible by 11)

5. Number: 639210

• 2: Yes (last digit is 0)
• 3: Yes (6 + 3 + 9 + 2 + 1 + 0 = 21, divisible by 3)
• 4: Yes (last two digits 10, not divisible by 4)
• 5: Yes (last digit is 0)
• 6: Yes (divisible by 2 and 3)
• 8: No (last three digits 210, not divisible by 8)
• 9: No (sum of digits 21, divisible by 9)
• 10: Yes (last digit is 0