Before we reach for a calculator or work through long division, we can often tell whether a number divides evenly just by looking at its digits.
Divisibility rules are quick shortcuts that let us check this in seconds, no scratch paper required.
To help you master these essential shortcuts, Mathnasium tutors will show you the rule for every number from 2 through 12, complete with simple examples for each one so you can spot patterns instantly and solve problems confidently.
A number is divisible by another if we can divide it evenly with no remainder left over.
When we divide two numbers, there are only two possible outcomes:
No remainder: 24 ÷ 6 = 4. Since it divides evenly with nothing left over, 24 is divisible by 6.
A remainder: 25 ÷ 6 = 4 with 1 left over. Since we have a leftover piece, 25 is not divisible by 6.
For small numbers, we can usually check this using basic multiplication facts. But as numbers grow larger, like 1,624 or 729, checking if a number is divisible usually means grabbing scratch paper and working through long division.
However, with divisibility rules, we have a way to skip the extra work and know the answer instantly just by looking at the digits.
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While divisibility tests can be created for many numbers, the most useful and commonly taught shortcuts are for 2 through 12.
Why?
First, 2 through 12 align with standard multiplication tables. These are the numbers we divide by most often when simplifying fractions, finding common denominators, or factoring.
Second, they are the “sweet spot” for mental math. For numbers beyond 12 (like 13 or 17), divisibility tests tend to get complex and clunky. In many cases, using quick long division is just as fast as following a multi-step rule.
So today, we’ll break down how 2 to 12 work with simple examples you can put to use right away.
A number is divisible by 2 if it ends in 0, 2, 4, 6, or 8. Notice something about those numbers?
They're all even—and yes, 0 is even too, because 0 ÷ 2 = 0 with no remainder!
So to check if any number is divisible by 2, all we have to do is glance at the very last digit.
Let's test 138. It ends in 8, which is even. So it should divide evenly by 2. Let's check: 138 ÷ 2 = 69. Yes! Works perfectly.
Now let's try 425. It ends in 5, which is odd. Since 5 isn't on our list, 425 won't divide evenly. (If we check: 425 ÷ 2 = 212 with a remainder of 1).
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A number is divisible by 3 if the sum of all its digits is divisible by 3.
Instead of doing long division on a big number, we can just add its digits together. If that total is in our 3s counting table, the whole number works!
Let's test it out:
Take 111. Add the digits: 1 + 1 + 1 = 3. Is 3 divisible by 3? Yes! So 111 should divide evenly. Let's check: 111 ÷ 3 = 37. Perfect!
Let's try a bigger one: 729. Add the digits: 7 + 2 + 9 = 18. Since 18 is divisible by 3, 729 works too! Let's check: 729 ÷ 3 = 243. Yes!
Now let's try 526. Add the digits: 5 + 2 + 6 = 13. Is 13 in our 3s counting table? Nope. So 526 won't divide evenly. (If we check the math: 526 ÷ 3 = 175 with a remainder of 1).
A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
Here’s why we find this shortcut so handy. 100 is always divisible by 4. That means no matter how huge a number gets, the hundreds, thousands, and millions places don't affect divisibility at all.
We can completely ignore the rest of the number and focus only on the last two digits!
To see this in action:
First, we’ll try 731. Look at the last two digits: 31. Is 31 in our 4s counting table? Nope. So 731 won't divide evenly. (Checking the math: 731 ÷ 4 = 182 with a remainder of 3).
Now, we’ll try 1,624. Focus on just the last two digits: 24. Is 24 divisible by 4? Yes, 24 ÷ 4 = 6! So 1,624 should divide evenly. Let's check: 1,624 ÷ 4 = 406. It works!
A number is divisible by 5 if its last digit is 0 or 5.
This is one of the easiest shortcuts to spot because counting by 5s always creates a strict, repeating end pattern (5, 10, 15, 20, 25...).
Take 835. Glance at the final digit: 5. Because it ends in 5, it lands right on our count! Let's check: 835 ÷ 5 = 167. Sure enough, no remainder!
Now, try 942. Look at the final digit: 2. Since 2 is neither 0 nor 5, 942 won't divide cleanly. (Checking the math once more: 942 ÷ 5 = 188 with a remainder of 2).
A number is divisible by 6 if it is divisible by both 2 and 3.
Since 6 is made by multiplying 2 and 3, a number has to pass both tests to work! That means it must be an even number (ends in 0, 2, 4, 6, or 8) and its digits must add up to a multiple of 3.
First, take 408.
Test for 2: It ends in 8 (even), so it passes!
Test for 3: Add the digits: 4 + 0 + 8 = 12. Since 12 is divisible by 3, it passes this test too! Because it passed both, 408 is divisible by 6. Let's confirm: 408 ÷ 6 = 68. Yes!
Now, try 734.
Test for 2: It ends in 4 (even), so it passes the first test.
Test for 3: Add the digits: 7 + 3 + 4 = 14. Is 14 divisible by 3? Nope. Since it failed the 3s test, 734 won't divide evenly by 6. (734 ÷ 6 = 122 with a remainder of 2).
A number is divisible by 7 if doubling the last digit and subtracting it from the remaining digits leaves a number divisible by 7 (or 0).
Yes, we know what you're thinking. This one is the most complex so far! Still, this two-step subtraction trick can spare us a lot of time compared to doing full long division on a big number.
Say we want to check if 343 is divisible by 7. We’ll do it step by step:
Separate the last digit (3) from the rest (34).
Double the last digit: 3 × 2 = 6.
Subtract from the rest: 34 - 6 = 28. Since 28 is divisible by 7 (7 × 4 = 28), 343 is too! Let's check this too: 343 ÷ 7 = 49.
Pretty neat, right?
Now, let’s do 255.
Separate the last digit (5) from the rest (25).
Double the last digit: 5 × 2 = 10.
Subtract from the rest: 25 - 10 = 15. Is 15 in our 7s counting table? Nope (7, 14, 21...). So 255 won't divide evenly. (255 ÷ 7 = 36 with a remainder of 3).
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A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
If this feels familiar, it's because it works just like the rule for 4! Since 1,000 is always divisible by 8, we can completely ignore the thousands, ten-thousands, and millions places. The only part that matters is the final three digits.
Consider 3,112. Look at just the last three digits: 112. Is 112 divisible by 8? Dividing gives 112 ÷ 8 = 14. Since that worked, 3,112 is guaranteed to divide evenly too! 3,112 ÷ 8 = 389).
On the flip side, test 5,214. Focus on the last three digits: 214. When we try dividing 214 by 8, we get 26 with a remainder of 6. Because 214 failed the test, 5,214 won't divide cleanly either. (5,214 ÷ 8 = 651 with a remainder of 6).
A number is divisible by 9 if the sum of its digits is divisible by 9.
It follows the exact same pattern as the rule for 3. We just swap in 9!
317: Add the digits: 3 + 1 + 7 = 11. Since 11 isn't in the 9s table, it fails. (317 ÷ 9 = 35 with a remainder of 2).
549: Add the digits: 5 + 4 + 9 = 18. Since 18 is divisible by 9, it passes! (549 ÷ 9 = 61).
A number is divisible by 10 if its last digit is 0.
This is easily the simplest shortcut of them all. No math or adding required!
850: Check the final digit: 0. That's all it takes! (850 ÷ 10 = 85).
643: Check the final digit: 3. Since it doesn't end in 0, it misses the mark. (643 ÷ 10 = 64 with a remainder of 3).
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A number is divisible by 11 if we add every other digit, subtract the two totals, and get 0 or a number in the 11s table.
Think of it like playing hopscotch across the number! We skip every second digit to make two separate teams, add each team up, and find the difference between them.
Here is how we test it:
For 825:
We jump across: 8 and 5 make one team (8 + 5 = 13).
The leftover digit, 2, is on the other team.
We subtract the teams: 13 - 2 = 11. Since 11 is in the 11s table, it passes! (825 ÷ 11 = 75).
Let's turn it up a notch and try a bigger number! Consider 2,816:
We make our first team: 2 and 1 (2 + 1 = 3).
We make our second team: 8 and 6 (8 + 6 = 14).
We subtract the smaller team from the bigger team: 14 - 3 = 11. Another hit! (We can use our calculators to check this: 2,816 ÷ 11 = 256).
A number is divisible by 12 if it is divisible by both 3 and 4.
Just like we saw with 6, a composite number's rule comes down to its building blocks! Since 3 × 4 = 12, a number has to pass both the 3s test (digits add up to a multiple of 3) and the 4s test (last two digits form a number divisible by 4).
So, is 648 divisible by 12? To check, we:
Test for 3: We add the digits: 6 + 4 + 8 = 18. Since 18 is divisible by 3, it passes!
Test for 4: We check the last two digits: 48. Since 48 ÷ 4 = 12, it passes here too! Because it passed both tests, 648 is divisible by 12. (Checking the math: 648 ÷ 12 = 54).
What about 514?
Test for 3: We add the digits: 5 + 1 + 4 = 10. Since 10 isn't divisible by 3, it fails right away!
We don't even need to check the 4s rule. (514 ÷ 12 = 42 with a remainder of 10).
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Learning divisibility rules helps students recognize number patterns and solve problems more efficiently.
The best way to remember divisibility rules is to practice applying them to different numbers.
Read each question below, decide whether the number is divisible by the given divisor, and then check your answers to see how you did.
Is 78 divisible by 2?
Is 415 divisible by 3?
Is 1,324 divisible by 4?
Is 94 divisible by 5?
Is 1,617 divisible by 7?
Is 837 divisible by 9?
Is 2,490 divisible by 10?
Is 93 divisible by 11?
Is 2,136 divisible by 12?
At Mathnasium, we believe strong division skills come from understanding why numbers behave the way they do, not just memorizing steps to follow.
Mathnasium is a math-only learning center dedicated to helping K-12 students of all skill levels excel in math.
Students come to us with different levels of familiarity with divisibility rules. Some are just learning how the rules work, while others are ready to apply them to more complex problems.
At Mathnasium, they develop the confidence to recognize patterns, test divisibility efficiently, and approach new math challenges with greater ease.
We build that foundation through the Mathnasium Method™, our proprietary teaching approach. This proven approach includes:
Assessment and Personalized Learning Plans: Each student begins with a diagnostic assessment to identify current skills, strengths, and gaps. From those findings, we build a personalized learning plan tailored to their goals, whether that means strengthening foundational skills, building fluency, or developing the reasoning flexibility that STEM careers require.
Teaching for Understanding: Our specially trained tutors use natural language and a mix of verbal, visual, mental, tactile, and written techniques so each concept lands before we move forward.
Problem-Solving and Critical Thinking: Our tutors know when to offer support and when to let students work through a problem on their own. That balance builds the independence and persistence that STEM careers reward.
An Engaging and Fun Learning Environment: Sessions include games, earned rewards, and consistent celebration of progress. Students learn to treat a difficult problem as a puzzle rather than a verdict on their ability.
The results speak for themselves:
94% of parents report improvement in their child's math skills and understanding
93% of parents report an improved attitude toward math after attending Mathnasium
90% of students saw improvement in their school grades
With over 1,100 learning centers across North America, there is likely a Mathnasium close to you.
Families across Logan, North Logan, Providence, Hyde Park, Nibley, Wellsville, River Heights, Smithfield, and Hyrum trust Mathnasium of Logan to help their children build math confidence.
Whether your child is building confidence with divisibility rules or needs extra support in math, our team is here to help.
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Yes (78 ends in 8, an even number)
No (4 + 1 + 5 = 10, and 10 is not divisible by 3)
Yes (The last two digits are 24, and 24 is divisible by 4)
No (Numbers divisible by 5 end in 0 or 5)
Yes (1,617 ÷ 7 = 231)
Yes (8 + 3 + 7 = 18, and 18 is divisible by 9)
Yes (It ends in 0)
No (93 ÷ 11 = 8 remainder 5, or using the 11 rule: (9 − 3) = 6, which is not a multiple of 11).
Yes (It is divisible by 3 (2 + 1 + 3 + 6 = 12) and 4 (last two digits are 36), so it is divisible by 12)
Mathnasium of Logan is a math-only learning center for K-12 students in Logan, UT. Trusted by over a million parents, Mathnasium uses personalized learning plans and the proprietary Mathnasium Method™ to help students catch up, keep up, and get ahead on their math journey.
Our specially trained tutors deliver face-to-face instruction in a supportive and fun small-group environment, working with students both in center and online to develop a deep understanding of math, build confidence, and improve academic performance.
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