How to Check Divisibility by 13 Using the Rule and Solved 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 Rules for 13 with Step by Step Method
1. What is the divisibility rule for 13?
The divisibility rule for 13 states that a number is divisible by 13 if repeatedly multiplying its last digit by 4 and adding it to the remaining number results in a multiple of 13.
- Step 1: Separate the last digit from the number.
- Step 2: Multiply the last digit by 4.
- Step 3: Add the result to the remaining truncated number.
- Step 4: Repeat until you get a small number.
2. How do you check if a number is divisible by 13?
To check if a number is divisible by 13, apply the rule of multiplying the last digit by 4 and adding it to the remaining number repeatedly.
- Example: Check 351.
- Last digit = 1 → 1 × 4 = 4
- Remaining number = 35
- Add: 35 + 4 = 39
3. Can you give an example of the divisibility rule of 13?
Yes, for example, the number 572 can be tested using the divisibility rule for 13.
- Last digit = 2 → 2 × 4 = 8
- Remaining number = 57
- Add: 57 + 8 = 65
4. Why does the divisibility rule for 13 work?
The divisibility rule for 13 works because it is based on modular arithmetic and the fact that 10 ≡ −3 (mod 13), which leads to the multiplier 4 in the shortcut method.
- Each step simplifies the number without changing its remainder when divided by 13.
- The process reduces large numbers to smaller equivalent values.
5. Is there an alternative rule for divisibility by 13?
Yes, an alternative divisibility rule for 13 is to multiply the last digit by 9 and subtract it from the remaining number.
- Step 1: Separate the last digit.
- Step 2: Multiply it by 9.
- Step 3: Subtract from the remaining number.
6. What are the first few multiples of 13?
The first few multiples of 13 are numbers obtained by multiplying 13 by whole numbers.
- 13 × 1 = 13
- 13 × 2 = 26
- 13 × 3 = 39
- 13 × 4 = 52
- 13 × 5 = 65
- 13 × 6 = 78
7. Is 1001 divisible by 13?
Yes, 1001 is divisible by 13 because applying the rule gives a multiple of 13.
- Last digit = 1 → 1 × 4 = 4
- Remaining number = 100
- Add: 100 + 4 = 104
- Repeat: 4 × 4 = 16; 10 + 16 = 26
8. What is the easiest way to remember the divisibility rule for 13?
The easiest way to remember the divisibility rule for 13 is the phrase: “Multiply the last digit by 4 and add.”
- Take the last digit.
- Multiply by 4.
- Add to the remaining number.
- Repeat until small.
9. How is the divisibility rule for 13 different from the rule for 11?
The divisibility rule for 13 uses multiplication and addition, while the rule for 11 uses alternating digit sums.
- Rule for 13: Multiply last digit by 4 and add to remaining number.
- Rule for 11: Find the difference between the sum of alternating digits.
10. Can the divisibility rule for 13 be used for very large numbers?
Yes, the divisibility rule for 13 works for very large numbers because the process reduces them step by step without changing divisibility.
- Apply the rule repeatedly.
- Each step makes the number smaller.
- Stop when you reach a manageable number.
