Пошаговые инструкции
Identify Your Integer
Begin by clearly identifying the integer you wish to test for divisibility. Write it down to avoid errors, especially with multi-digit numbers.
Review Divisibility Rules for Each Potential Divisor (2-13)
Familiarize yourself with the specific rule for each number from 2 to 13. Understand the logic behind each rule, paying close attention to the special procedures for 7 and 13.
Apply Each Rule Systematically
For each divisor (2 through 13), apply its corresponding rule to your chosen integer. Perform the necessary arithmetic (summing digits, checking last digits, performing alternating sums, or iterative steps for 7 and 13). Keep your calculations organized.
Record and Interpret Your Results
After applying each rule, note whether the integer is divisible by that specific number. Create a clear list of all the divisors that apply to your integer. For rules that yield a 'YES' or 'NO', clearly mark your findings.
Verify Complex Rules (Optional, but Recommended)
For rules like 7 and 13, which involve iterative steps, if you are unsure of your result or the number is very large, perform a direct division on the final, smaller number obtained from the rule to confirm its divisibility by 7 or 13. This helps prevent errors from repeated calculations.
Divisibility rules are powerful shortcuts that allow you to determine if one integer can be divided by another without leaving a remainder, often without performing long division. Mastering these rules enhances your number sense, simplifies factorization, and aids in various mathematical and real-world calculations.
Understanding Divisibility Rules
Prerequisites
To effectively apply these rules, you should have a basic understanding of:
- Integers: Whole numbers (positive, negative, and zero).
- Basic Arithmetic: Addition, subtraction, multiplication, and division.
- Place Value: Understanding digits' positions in a number (units, tens, hundreds, etc.).
The Core Concept
An integer 'A' is divisible by an integer 'B' if, when 'A' is divided by 'B', the remainder is zero. For example, 10 is divisible by 5 because 10 ÷ 5 = 2 with no remainder. However, 10 is not divisible by 3 because 10 ÷ 3 = 3 with a remainder of 1.
Step-by-Step Divisibility Checks
Here are the rules for checking divisibility from 2 to 13:
Rule for 2
A number is divisible by 2 if its last digit is an even number (0, 2, 4, 6, or 8).
- Example: 4,780 is divisible by 2 because its last digit is 0.
Rule for 3
A number is divisible by 3 if the sum of its digits is divisible by 3.
- Example: 4,788 is divisible by 3 because 4 + 7 + 8 + 8 = 27, and 27 is divisible by 3 (27 ÷ 3 = 9).
Rule for 4
A number is divisible by 4 if the number formed by its last two digits is divisible by 4. If the number has only two digits, check if that number itself is divisible by 4.
- Example: 4,788 is divisible by 4 because the number formed by its last two digits, 88, is divisible by 4 (88 ÷ 4 = 22).
Rule for 5
A number is divisible by 5 if its last digit is 0 or 5.
- Example: 4,780 is divisible by 5 because its last digit is 0.
Rule for 6
A number is divisible by 6 if it is divisible by both 2 and 3. This means its last digit must be even, and the sum of its digits must be divisible by 3.
- Example: 4,788 is divisible by 6 because it's divisible by 2 (last digit 8 is even) and by 3 (sum of digits 27 is divisible by 3).
Rule for 7
This rule is iterative. Take the last digit, double it, and subtract this result from the number formed by the remaining digits. If the final result is 0 or divisible by 7, the original number is divisible by 7. Repeat if necessary.
- Formula/Procedure: For a number
10a + b, it's divisible by 7 ifa - 2bis divisible by 7. - Example: Is 343 divisible by 7?
a=34,b=3.34 - (2 * 3) = 34 - 6 = 28. Since 28 is divisible by 7, 343 is divisible by 7.
Rule for 8
A number is divisible by 8 if the number formed by its last three digits is divisible by 8. For numbers with fewer than three digits, check if the number itself is divisible by 8.
- Example: 4,784 is divisible by 8 because the number formed by its last three digits, 784, is divisible by 8 (784 ÷ 8 = 98).
Rule for 9
A number is divisible by 9 if the sum of its digits is divisible by 9.
- Example: 4,788 is not divisible by 9 because 4 + 7 + 8 + 8 = 27, and 27 is divisible by 9 (27 ÷ 9 = 3). Correction: 4,788 IS divisible by 9. My example was for 4,788, not 4,784. Let's use 4,788 for the example. So, 4,788 is divisible by 9 because 27 is divisible by 9.
Rule for 10
A number is divisible by 10 if its last digit is 0.
- Example: 4,780 is divisible by 10 because its last digit is 0.
Rule for 11
Find the alternating sum of the digits (subtract the second digit from the first, add the third, subtract the fourth, and so on). If this result is 0 or divisible by 11, the original number is divisible by 11.
- Example: Is 4,785 divisible by 11?
4 - 7 + 8 - 5 = 0. Since the result is 0, 4,785 is divisible by 11.
Rule for 12
A number is divisible by 12 if it is divisible by both 3 and 4.
- Example: 4,788 is divisible by 12 because it's divisible by 3 (sum of digits 27) and by 4 (last two digits 88).
Rule for 13
This rule is also iterative. Take the last digit, multiply it by 4, and add it to the number formed by the remaining digits. If the final result is 0 or divisible by 13, the original number is divisible by 13. Repeat if necessary.
- Formula/Procedure: For a number
10a + b, it's divisible by 13 ifa + 4bis divisible by 13. - Example: Is 845 divisible by 13?
a=84,b=5.84 + (4 * 5) = 84 + 20 = 104. Is 104 divisible by 13?a=10,b=4.10 + (4 * 4) = 10 + 16 = 26. Since 26 is divisible by 13, 845 is divisible by 13.
Worked Example: Checking Divisibility of 47,880
Let's apply the rules to the number 47,880:
- By 2: Last digit is 0 (even). YES.
- By 3: Sum of digits = 4 + 7 + 8 + 8 + 0 = 27. 27 is divisible by 3. YES.
- By 4: Last two digits form 80. 80 is divisible by 4 (80 ÷ 4 = 20). YES.
- By 5: Last digit is 0. YES.
- By 6: Divisible by both 2 and 3. YES.
- By 7:
4788 - (2 * 0) = 4788. This does not simplify it much. Let's try478 - (2 * 8) = 478 - 16 = 462. Then46 - (2 * 2) = 46 - 4 = 42. 42 is divisible by 7. YES. - By 8: Last three digits form 880. 880 is divisible by 8 (880 ÷ 8 = 110). YES.
- By 9: Sum of digits = 27. 27 is divisible by 9. YES.
- By 10: Last digit is 0. YES.
- By 11: Alternating sum: 4 - 7 + 8 - 8 + 0 = -3. -3 is not 0 or divisible by 11. NO.
- By 12: Divisible by both 3 and 4. YES.
- By 13:
4788 + (4 * 0) = 4788. This does not simplify it much. Let's try478 + (4 * 8) = 478 + 32 = 510. Then51 + (4 * 0) = 51. 51 is not divisible by 13 (13 * 3 = 39, 13 * 4 = 52). NO.
Therefore, 47,880 is divisible by 2, 3, 4, 5, 6, 7, 8, 9, 10, and 12, but not by 11 or 13.
Common Pitfalls to Avoid
- Misapplying Rules: Ensure you are using the correct rule for each divisor. For instance, confusing the rule for 3 with the rule for 9.
- Calculation Errors: Simple addition or subtraction mistakes when summing digits or performing the iterative steps for rules 7 and 13 can lead to incorrect results.
- Incomplete Checks: For composite divisors like 6, 10, or 12, remember to check all their prime factors (e.g., for 6, check both 2 and 3).
- Large Numbers with Complex Rules: For rules 7 and 13, repeatedly applying the steps to very large numbers can be cumbersome and error-prone. Sometimes direct division for the final, smaller number is faster than repeating the rule.
When to Use a Divisibility Calculator
While manual checks are excellent for understanding and for smaller numbers, a divisibility calculator offers significant advantages for:
- Very Large Numbers: When the integer has many digits, summing digits or performing iterative subtractions/additions can be time-consuming and tedious.
- Speed and Efficiency: For quick checks in a professional context or when dealing with multiple numbers rapidly.
- Verification: To double-check your manual calculations and ensure accuracy, especially for critical tasks.
- Learning Aid: To instantly see the results and compare them with your manual attempts, reinforcing your understanding.
By mastering these manual divisibility rules, you gain a foundational understanding of number theory. For convenience and speed with larger numbers, a digital calculator can be a valuable complementary tool.