You are in the middle of a SASMO practice test. You see a number theory question that asks: “How many integers between 100 and 500 are divisible by 6 but not by 9?” Your first instinct might be to start dividing each number. That would take forever. There is a better way.

Divisibility rules let you check whether one number divides another without doing long division. They are shortcuts that turn a messy calculation into a simple check. For SASMO, AMC, and other math competitions, these rules are not just nice to know. They are essential. They save you time, reduce errors, and help you spot patterns that lead to the answer faster.

Why Divisibility Rules Matter for SASMO

SASMO number theory problems often ask about factors, multiples, remainders, and prime numbers. A typical question might ask: “What is the smallest number that is divisible by 2, 3, and 5?” Without rules, you would test numbers one by one. With rules, you know the answer must end in 0 (rule for 2 and 5) and have a digit sum divisible by 3 (rule for 3). The smallest such number is 30.

These rules also help with more complex problems. For example, when you need to find the greatest common divisor (GCD) or least common multiple (LCM) of large numbers, divisibility checks help you factor them quickly. They are the foundation of many other number theory techniques.

The 7 Must-Know Divisibility Rules

Here are the rules that appear most often in SASMO and similar contests. Learn them, practice them, and they will become second nature.

Rule for 2: Check the Last Digit

A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).

• Example: 1,234 ends in 4. It is divisible by 2.
• Example: 1,235 ends in 5. It is not divisible by 2.

This rule is simple, but it is the building block for other rules.

Rule for 3: Sum the Digits

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

• Example: 1,236. Sum = 1 + 2 + 3 + 6 = 12. 12 is divisible by 3. So 1,236 is divisible by 3.
• Example: 1,237. Sum = 1 + 2 + 3 + 7 = 13. 13 is not divisible by 3. So 1,237 is not divisible by 3.

You can repeat the sum if you get a large number. For 987,654, sum = 9+8+7+6+5+4 = 39. Then 3+9 = 12. 12 is divisible by 3.

Rule for 4: Check the Last Two Digits

A number is divisible by 4 if the last two digits form a number divisible by 4.

• Example: 5,724. Last two digits are 24. 24 is divisible by 4. So 5,724 is divisible by 4.
• Example: 5,723. Last two digits are 23. 23 is not divisible by 4.

This rule is useful because you ignore everything except the last two digits.

Rule for 5: Check the Last Digit

A number is divisible by 5 if its last digit is 0 or 5.

• Example: 1,235 ends in 5. It is divisible by 5.
• Example: 1,236 ends in 6. It is not divisible by 5.

Combine this with the rule for 2. If a number ends in 0, it is divisible by both 2 and 5.

Rule for 6: Combine Rules for 2 and 3

A number is divisible by 6 if it is divisible by both 2 and 3. That means it must be even, and its digit sum must be divisible by 3.

• Example: 1,236. It is even (ends in 6). Digit sum = 12, which is divisible by 3. So 1,236 is divisible by 6.
• Example: 1,237. It is odd. So it is not divisible by 6, regardless of the digit sum.

Rule for 7: Double and Subtract

This rule is a bit more involved, but it is powerful. For a number, double the last digit and subtract it from the rest of the number. Repeat until you get a small number that you can check.

• Example: 1,274. Last digit is 4. Double it = 8. Subtract from the rest: 127 – 8 = 119. Now repeat with 119. Last digit is 9. Double it = 18. Subtract: 11 – 18 = -7. The absolute value is 7. 7 is divisible by 7. So 1,274 is divisible by 7.

This rule works for any number. Practice it a few times, and it will become automatic.

Rule for 8: Check the Last Three Digits

A number is divisible by 8 if the last three digits form a number divisible by 8.

• Example: 9,128. Last three digits are 128. 128 divided by 8 is 16. So 9,128 is divisible by 8.
• Example: 9,129. Last three digits are 129. 129 divided by 8 is not a whole number.

Rule for 9: Sum the Digits (Like 3)

A number is divisible by 9 if the sum of its digits is divisible by 9.

• Example: 1,233. Sum = 1 + 2 + 3 + 3 = 9. 9 is divisible by 9. So 1,233 is divisible by 9.
• Example: 1,234. Sum = 1 + 2 + 3 + 4 = 10. 10 is not divisible by 9.

Rule for 10: Check the Last Digit

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

• Example: 1,230 ends in 0. It is divisible by 10.
• Example: 1,231 ends in 1. It is not divisible by 10.

Rule for 11: Alternating Sum

A number is divisible by 11 if the alternating sum of its digits (starting from the left with a plus sign) is divisible by 11.

• Example: 1,331. Alternating sum: +1 – 3 + 3 – 1 = 0. 0 is divisible by 11. So 1,331 is divisible by 11.
• Example: 1,332. Alternating sum: +1 – 3 + 3 – 2 = -1. -1 is not divisible by 11.

Rule for 13: Multiply and Add (Similar to 7)

This rule is similar to the rule for 7, but you multiply the last digit by 4 and add it to the rest of the number. Repeat until you get a small number.

• Example: 1,274. Last digit is 4. Multiply by 4 = 16. Add to the rest: 127 + 16 = 143. Now repeat with 143. Last digit is 3. Multiply by 4 = 12. Add: 14 + 12 = 26. 26 is divisible by 13. So 1,274 is divisible by 13.

Comparison Table: Divisibility Rules

Divisor	Rule	Example	Common Mistake
2	Last digit is even	1,234 is divisible by 2	Forgetting that 0 is even
3	Sum of digits divisible by 3	1,236 is divisible by 3	Only checking the last digit
4	Last two digits divisible by 4	5,724 is divisible by 4	Forgetting to check two digits
5	Last digit is 0 or 5	1,235 is divisible by 5	Thinking any number ending in 5 is divisible by 5 (it is)
6	Divisible by 2 and 3	1,236 is divisible by 6	Checking only one condition
7	Double last digit, subtract from rest	1,274 is divisible by 7	Forgetting to repeat the process
8	Last three digits divisible by 8	9,128 is divisible by 8	Only checking the last digit
9	Sum of digits divisible by 9	1,233 is divisible by 9	Confusing with rule for 3
10	Last digit is 0	1,230 is divisible by 10	Thinking 5 works
11	Alternating sum divisible by 11	1,331 is divisible by 11	Starting with the wrong sign
13	Multiply last digit by 4, add to rest	1,274 is divisible by 13	Using the rule for 7 instead

Common Mistakes and How to Avoid Them

Even experienced students make errors with divisibility rules. Here are the most frequent ones:

• Forgetting to repeat the process for 7 and 13. You cannot stop after one step. Keep going until you get a small number.
• Mixing up rules for 3 and 9. Both use digit sums, but 9 requires the sum to be divisible by 9, not just 3.
• Only checking the last digit for 4 and 8. You must check the last two digits for 4 and the last three for 8.
• Thinking that a number divisible by 2 and 5 is divisible by 10. It is, but that is because 10 = 2 * 5. For 6, you need 2 and 3.

Expert Advice: When you are unsure, write down the rule on scratch paper. A quick note like “7: double last, subtract” can prevent a silly mistake. Practice each rule with random numbers until you can apply it without thinking.

How to Practice Divisibility Rules for SASMO

Follow this process to build speed and accuracy:

1. Learn one rule at a time. Start with 2, 3, 5, and 10. Master those before moving to 4, 6, 8, and 9. Then tackle 7, 11, and 13.
2. Create a set of practice numbers. Write down 20 random numbers between 100 and 10,000. For each number, check divisibility by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 13. Time yourself.
3. Use flashcards. Write the divisor on one side and the rule on the other. Shuffle them and test yourself.
4. Apply rules to SASMO practice problems. Many SASMO number theory questions require you to combine rules. For example, a problem might ask for the smallest number divisible by 2, 3, and 5. Use the rules to find 30 quickly.
5. Review your mistakes. When you get a question wrong, check which rule you misapplied. Fix that rule in your memory.

For more practice, check out our number theory practice problems designed for SASMO competitors. You can also use our grade-by-grade problem sets to find the right level for you.

A Step-by-Step Walkthrough

Let us solve a SASMO-style problem using these rules.

Problem: How many integers between 100 and 500 are divisible by 6 but not by 9?

Step 1: Understand the conditions. A number divisible by 6 must be divisible by 2 and 3. A number divisible by 9 must have a digit sum divisible by 9.

Step 2: Find numbers divisible by 6. They are even and have a digit sum divisible by 3.

Step 3: From those, remove numbers divisible by 9. A number divisible by 9 is also divisible by 3, but we need it to be divisible by 6 (even) and not by 9.

Step 4: List the numbers divisible by 6 between 100 and 500. The first is 102 (even, digit sum 3). The last is 498 (even, digit sum 21, divisible by 3). The count is (498 – 102) / 6 + 1 = 67.

Step 5: Now remove numbers divisible by 18 (since a number divisible by 6 and 9 is divisible by LCM(6,9) = 18). The first divisible by 18 is 108. The last is 486. Count = (486 – 108) / 18 + 1 = 22.

Step 6: Subtract: 67 – 22 = 45.

The answer is 45. Without divisibility rules, you would have to check each number individually. With rules, you solve it in under a minute.

When to Use Each Rule in a SASMO Test

Not every problem needs every rule. Here is a guide:

• If the problem mentions “even” or “odd,” use the rule for 2.
• If the problem asks about multiples of 3 or 9, use digit sums.
• If the problem involves factors of 4 or 8, check the last two or three digits.
• If the problem mentions 6, 12, or 15, combine rules for their prime factors.
• If the problem involves 7, 11, or 13, use the special rules.

For a deeper understanding of how these rules connect to other math topics, read our guide on why number theory is the secret weapon every SASMO competitor needs. You can also learn about modular arithmetic, which is the mathematical foundation behind these rules.

Building Your Divisibility Rules Toolkit

Here is a bulleted list of the rules for quick reference:

• 2: Last digit is even.
• 3: Sum of digits divisible by 3.
• 4: Last two digits divisible by 4.
• 5: Last digit is 0 or 5.
• 6: Divisible by 2 and 3.
• 7: Double last digit, subtract from rest, repeat.
• 8: Last three digits divisible by 8.
• 9: Sum of digits divisible by 9.
• 10: Last digit is 0.
• 11: Alternating sum of digits divisible by 11.
• 13: Multiply last digit by 4, add to rest, repeat.

Print this list and keep it near your study area. Use it until you no longer need it.

Your Next Steps for SASMO Success

Divisibility rules are one piece of the puzzle. To perform well on SASMO, you also need strong skills in other areas. Check out these resources:

• How to build strong algebraic thinking for math olympiads
• Combinatorics made simple: counting principles for SASMO success
• 5 mental math shortcuts every SASMO competitor should master
• Building your SASMO competition toolkit: essential formulas and patterns to memorize
• How to manage your time effectively during SASMO competition day

Turn These Rules Into Automatic Habits

The difference between a good SASMO score and a great one often comes down to speed. When you can apply a divisibility rule in two seconds instead of ten, you save precious time for harder problems. That time adds up across a whole test.

Start today. Pick one rule you do not know well. Practice it for five minutes. Tomorrow, add a second rule. By the end of the week, you will have all 11 rules in your toolbox. By the end of the month, they will be automatic.

You have everything you need to master these rules. Now go practice. Your next SASMO test will thank you.