Divisibility Rule For 1 To 10

Divisibility rules are shortcuts that help you determine if one number can be divided by another without leaving a remainder. They are fundamental tools in mathematics, simplifying tasks from basic arithmetic to complex problem-solving in algebra and number theory. In practice, mastering these rules for numbers 1 through 10 builds a strong foundation for numerical fluency and confidence. This practical guide will walk you through each rule, explain the underlying logic, and provide clear examples.

Worth pausing on this one.

Divisibility Rule for 1

This is the simplest rule of all. Every integer is divisible by 1. Since dividing any number by 1 always yields the same number, there is never a remainder.

• Example: 7 is divisible by 1 (7 ÷ 1 = 7). 125 is divisible by 1 (125 ÷ 1 = 125). Even a large number like 9,482 is divisible by 1.

Divisibility Rule for 2

A number is divisible by 2 if its last digit is even. The even digits are 0, 2, 4, 6, and 8.

• Example: 342 is divisible by 2 because the last digit is 2. 1,567 is not divisible by 2 because the last digit is 7 (odd).

Divisibility Rule for 3

A number is divisible by 3 if the sum of its digits is divisible by 3.

• Example: Let's check 483.
  1. Add the digits: 4 + 8 + 3 = 15.
  2. Is 15 divisible by 3? Yes, 15 ÷ 3 = 5.
  3. So, 483 is divisible by 3 (483 ÷ 3 = 161).

Divisibility Rule for 4

A number is divisible by 4 if the number formed by its last two digits is divisible by 4.

• Example: Let's check 10,324.
  1. Look at the last two digits: 24.
  2. Is 24 divisible by 4? Yes, 24 ÷ 4 = 6.
  3. So, 10,324 is divisible by 4 (10,324 ÷ 4 = 2,581).
  • Why it works: Any number can be expressed as 100 × (the number formed by all but the last two digits) + (the last two digits). Since 100 is divisible by 4 (100 ÷ 4 = 25), the entire number is divisible by 4 if and only if the last two digits are.

Divisibility Rule for 5

A number is divisible by 5 if its last digit is either 0 or 5 Worth keeping that in mind..

• Example: 790 is divisible by 5 (ends in 0). 35 is divisible by 5 (ends in 5). 123 is not divisible by 5 (ends in 3).

Divisibility Rule for 6

A number is divisible by 6 if it is divisible by both 2 and 3. You can apply the rules for 2 and 3 sequentially.

• Example: Let's check 2,556.
  1. Divisible by 2? Yes, because it ends in 6 (even).
  2. Divisible by 3? Sum the digits: 2 + 5 + 5 + 6 = 18. Is 18 divisible by 3? Yes.
  3. Since it passes both tests, 2,556 is divisible by 6 (2,556 ÷ 6 = 426).

Divisibility Rule for 7

The rule for 7 is more complex and has several versions. One of the most common methods is as follows:

1. Double the last digit of the number.
2. Subtract this doubled value from the rest of the number (the number without its last digit).
3. Repeat the process on the new, smaller number until you get a number you recognize as being divisible by 7 or not.

• Example: Let's check 161.
  1. Double the last digit (1): 1 × 2 = 2.
  2. Subtract from the rest of the number: 16 - 2 = 14.
  3. Is 14 divisible by 7? Yes, 14 ÷ 7 = 2.
  4. That's why, 161 is divisible by 7 (161 ÷ 7 = 23).

Divisibility Rule for 8

A number is divisible by 8 if the number formed by its last three digits is divisible by 8.

• Example: Let's check 11,392.
  1. Look at the last three digits: 392.
  2. Is 392 divisible by 8? 392 ÷ 8 = 49. Yes.
  3. Because of this, 11,392 is divisible by 8 (11,392 ÷ 8 = 1,424).
  • Why it works: The logic is similar to the rule for 4. Any number can be expressed as 1,000 × (the number formed by all but the last three digits) + (the last three digits). Since 1,000 is divisible by 8 (1,000 ÷ 8 = 125), the entire number is divisible by 8 if the last three digits are.

Divisibility Rule for 9

A number is divisible by 9 if the sum of its digits is divisible by 9. This is very similar to the rule for 3.

• Example: Let's check 7,389.
  1. Add the digits: 7 + 3 + 8 + 9 = 27.
  2. Is 27 divisible by 9? Yes, 27 ÷ 9 = 3.
  3. That's why, 7,389 is divisible by 9 (7,389 ÷ 9 = 821).

Divisibility Rule for 10

A number is divisible by 10 if its last digit is 0 Worth knowing..

• Example: 50 is divisible by 10. 1,230 is divisible by 10. 45 is not divisible by 10.

Putting It All Together: A Practical Example

Let's use the number 2,520 to test multiple rules.

• Divisible by 2? Yes, ends in 0.
• Divisible by 3? Sum of digits: 2+5+2+0 = 9. Yes.
• Divisible by 4? Last

three digits: 520. Yes, divisible by 9 (2,520 ÷ 9 = 280). That said, 520 ÷ 8 = 65. ** Yes, ends in 0. This leads to * **Divisible by 7? Plus, yes. Also, * **Divisible by 10? ** Since it is divisible by both 2 and 3, yes. (2,520 ÷ 6 = 420). Think about it: * **Divisible by 6? ** Apply the rule: Double the last digit (0 × 2 = 0), subtract from the rest: 252 - 0 = 252. * **Divisible by 9?Now, check 252: double the last digit (2 × 2 = 4), subtract from 25: 25 - 4 = 21. Also, yes, 21 ÷ 7 = 3. Now, ** Last three digits: 520. Is 520 divisible by 4? That said, * **Divisible by 8? In practice, is 520 divisible by 8? Consider this: ** Sum of digits: 2+5+2+0 = 9. So, 2,520 is divisible by 7 (2,520 ÷ 7 = 360).

• **Divisible by 5?Which means yes. Also, is 21 divisible by 7? 520 ÷ 4 = 130. ** Yes, ends in 0.

As we can see, 2,520 is divisible by many numbers, which highlights its versatility in mathematical contexts like finding common denominators or simplifying ratios Not complicated — just consistent..

Conclusion

Mastering these divisibility rules is a valuable skill that enhances numerical fluency and efficiency. Whether you're checking calculations, factoring numbers, or working with fractions, these shortcuts save time and reduce errors. While the rules for 7 and 8 require a bit more practice, they are worth learning for their utility in mental math. By integrating these techniques into your routine, you can approach problems with greater confidence and speed, laying a strong foundation for more advanced mathematical concepts.

Beyond the basics for 2 through 10, several additional shortcuts prove handy when dealing with larger numbers or when you need to verify divisibility quickly without a calculator And it works..

Divisibility by 11
Form the alternating sum of the digits: subtract the second digit from the first, add the third, subtract the fourth, and so on. If the result is 0 or a multiple of 11, the original number is divisible by 11.
Example: 27 28 → (2 − 7 + 2 − 8) = −11, which is a multiple of 11, so 27 28 ÷ 11 = 2 480.

Divisibility by 12
A number is divisible by 12 exactly when it passes both the 3‑rule and the 4‑rule (since 12 = 3 × 4 and the factors are coprime). Check the digit sum for 3 and the last two digits for 4.
Example: 1 452 → digit sum = 12 (divisible by 3); last two digits = 52 (52 ÷ 4 = 13). Hence 1 452 is divisible by 12 And that's really what it comes down to. But it adds up..

Divisibility by 15
Because 15 = 3 × 5, a number must end in 0 or 5 (rule for 5) and have a digit sum divisible by 3.
Example: 3 465 ends in 5 and its digit sum 3+4+6+5 = 18 (divisible by 3), so it’s divisible by 15.

Divisibility by 18
Since 18 = 2 × 9, the number must be even (last digit 0,2,4,6,8) and its digit sum must be a multiple of 9.
Example: 5 832 is even and its digit sum 5+8+3+2 = 18 → divisible by 18.

Applying the Rules in Problem Solving
These shortcuts shine when simplifying fractions, finding least common multiples, or checking work in long division. To give you an idea, to reduce the fraction  462 /  693 , notice that both numbers are divisible by 3 (digit sums 12 and 21) and also by 7 (using the 7‑rule). Dividing numerator and denominator by 21 yields 22 / 33, which further reduces to 2 / 3 after another factor of 11.

Tips for Mastery

1. Practice with random numbers – pick a four‑digit integer and run through each rule mentally; the more you do it, the faster the patterns become intuitive.
2. Combine rules – remember that composite divisors often factor into smaller, coprime parts (e.g., 18 = 2 × 9, 20 = 4 × 5). Test each factor separately.
3. Use visual aids – writing the alternating sum for 11 or highlighting the last two/three digits helps avoid slips, especially under time pressure.
4. Check your work – after applying a rule, verify with a quick multiplication or division to reinforce confidence.

By extending your toolkit beyond the single‑digit divisors, you gain a versatile