How to Check if a Number is Divisible by 13: Stepwise Method & Examples
The concept of Divisibility Rules for 13 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. This rule helps students quickly check if a number can be divided by 13 without using direct division, saving time in calculations and competitive exams.
What Is the Divisibility Rule for 13?
A divisibility rule for 13 is a simple mathematical shortcut that lets you test whether a number is exactly divisible by 13. You’ll find this concept applied in topics such as number theory, quick calculation tricks, and factorization. The rule avoids long division, making it handy during timed quizzes, Olympiads, and exams like JEE or NTSE.
Key Formula for Divisibility by 13
Here’s the standard formula:
Remove the last digit from the number, multiply that digit by 9, and add the result to the truncated number. If the answer is divisible by 13 (including 0), the original number is divisible by 13.
Cross-Disciplinary Usage
Divisibility rules for 13 are not only useful in Maths but also help in Physics equations, coding (for checking cycles or periodicity), and logical puzzles that require pattern recognition. Students preparing for JEE or NEET will see its relevance when simplifying large numerical expressions.
Step-by-Step Illustration
- Let’s take the number 858.
Step 1: Remove the last digit (8). Remaining number = 85.Step 2: Multiply the last digit by 9 → 8 × 9 = 72.Step 3: Add 72 to the remaining number 85 → 85 + 72 = 157.Step 4: Repeat the process if needed: Remove last digit: 7, remaining: 15; 7 × 9 = 63; 15 + 63 = 78.Since 78 is divisible by 13 (13 × 6), the original number 858 is divisible by 13.
Alternative Rules for 13 (Quick Table)
| Rule | Description | Example (for 286) |
|---|---|---|
| Rule 1 | Multiply last digit by 9, add to rest | 28 + (6 × 9) = 28 + 54 = 82 (repeat: 8 + 2 × 9 = 26, which is 2 × 13) |
| Rule 2 | Multiply last digit by 4, add to rest | 28 + (6 × 4) = 28 + 24 = 52 (52/13 = 4) |
| Rule 3 | Alternating sum of 3-digit groups | (for larger numbers, like 2,453,674: 674 - 453 + 2 = 223) |
| Rule 4 | Subtract last two digits from 4 × other digits | (28 × 4) - 6 = 112 - 6 = 106 |
Speed Trick or Vedic Shortcut
Here’s a quick shortcut for the divisibility rule for 13 many students use during exams:
- Take the last digit of the number.
- Multiply it by 9 and add to the remaining number (as per the rule).
- If the result is big, repeat the steps until you get a small number.
- If that small number is divisible by 13, so is your starting number!
These tricks work in NTSE, Olympiads, and fast classroom quizzes. Vedantu’s live classes share more such mental math techniques for all divisibility rules.
Try These Yourself
- Check if 390 is divisible by 13.
- Is 728 divisible by 13?
- Find out if 1690 passes the divisibility rule for 13.
- Name three 3-digit numbers divisible by 13.
Frequent Errors and Misunderstandings
- Forgetting to repeat the process until a small, checkable number.
- Multiplying by the wrong number (it’s 9 for trick 1, not 13 itself).
- Applying the rule for divisibility by 3 or 11 instead of 13 by mistake.
Relation to Other Concepts
The idea of divisibility rules for 13 connects with factors, multiples, prime numbers, and the general divisibility rules for Maths. Mastering this helps with finding HCF, LCM, and solving number-based puzzles.
Classroom Tip
A quick way to remember: “Multiply the last digit by 9, add to the rest, check again!” Repeat until you see a familiar multiple of 13. Ask your teacher for a number chain or table to help in practice. Vedantu’s classes include printable tables for all tricky divisibility rules.
Summary Table: Rule, Example, Shortcut
| Rule Name | Process | Example Number | Result |
|---|---|---|---|
| Multiply last digit by 9, add | Truncate last digit, multiply by 9, add | 858 | 858 → 85+72=157 → 15+63=78 → 78/13=6 |
| Multiply last digit by 4, add | Truncate last digit, multiply by 4, add | 650 | 65+0=65 (divisible by 13: 5) |
| Subtract last two digits from 4×rest | (Other digits)×4 - (last two digits) | 728 | 7×4=28, 28-28=0; 0 is divisible by 13 |
| Alternating sum, blocks of three | Sum 3-digit blocks alternately | 2,453,674 | 674-453+2=223 (not divisible by 13) |
We explored Divisibility Rules for 13—from definition, formulas, examples, common mistakes, and how it links to other Maths topics like factors, multiples, and primes. Continue practicing with Vedantu to become quick and confident with divisibility skills for all numbers.
Divisibility Rules Full Guide | Factors of 13 | Multiples of 13 | Practice Divisibility Questions
FAQs on Divisibility Rule for 13: Shortcut, Examples, and Practice
1. What is the divisibility rule for 13 in Maths?
The divisibility rule for 13 involves a series of steps to determine if a number is divisible by 13 without performing long division. There are several variations of the rule; one common method is to multiply the last digit by 9, subtract the result from the remaining number (excluding the last digit), and check if the new number is divisible by 13. If it is, the original number is also divisible by 13. Other methods involve manipulating blocks of digits.
2. How do you quickly check if a number is divisible by 13?
The quickest method depends on the size of the number. For smaller numbers, directly applying the rule (multiplying the last digit by 9 and subtracting) is efficient. For larger numbers, alternative methods involving blocks of three digits might be faster. Practice with various methods to determine which works best for you.
3. Can you give an example of the divisibility test for 13?
Let's check if 286 is divisible by 13. Multiply the last digit (6) by 9: 6 * 9 = 54. Subtract this from the remaining number (28): 28 - 54 = -26. Since -26 is divisible by 13 (-26 ÷ 13 = -2), 286 is divisible by 13.
4. Why is it important to learn divisibility rules for 13, 17, or 19?
Learning divisibility rules for numbers like 13, 17, and 19 helps improve your understanding of number properties and provides efficient mental calculation shortcuts. This is particularly useful in competitive exams where speed and accuracy are crucial. They also aid in factorization and problem-solving in number theory.
5. Is the divisibility rule for 13 useful in competitive exams?
Absolutely! In competitive exams, time is of the essence. Knowing the divisibility rule for 13 allows you to quickly eliminate options and solve problems faster, saving valuable time and improving your overall score. It can be a significant advantage in situations where calculations need to be performed quickly.
6. What are the different methods for checking divisibility by 13?
There are several methods. One common approach involves subtracting nine times the last digit from the remaining digits. Another involves working with blocks of three digits, alternately adding and subtracting. The best method depends on the specific number and personal preference. Practice each method to find what works best.
7. How does the divisibility rule for 13 work with larger numbers?
For larger numbers, the process might need to be repeated. Using the method of subtracting nine times the last digit, you may need to repeat this process on the result until you get a small enough number to easily determine divisibility by 13.
8. What are some common mistakes to avoid when applying the divisibility rule for 13?
Common mistakes include incorrect multiplication or subtraction, forgetting to check the final result for divisibility by 13, and misinterpreting the rules. Careful attention to detail throughout the process is key.
9. How can I improve my speed and accuracy in applying the divisibility rule for 13?
Practice is key! Work through numerous examples, focusing on both small and large numbers. Regular practice will improve your speed and reduce the chance of errors. Try different methods to find what you find most intuitive.
10. Are there any alternative or easier ways to check for divisibility by 13 besides the standard rule?
While the standard rule is widely used, some might find alternative methods more intuitive. These could include using a calculator or programming a simple algorithm to check divisibility more efficiently. However, understanding the underlying principle remains crucial.
11. How does the divisibility rule for 13 relate to other divisibility rules?
Understanding the divisibility rule for 13 can enhance your understanding of other divisibility rules for prime numbers. The principles often involve manipulating digits in a systematic way, and mastering one can improve your skills with others.
