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SSAT Middle Level: Quantitative • Number Properties and Integers

Use divisibility rules to test a number.

Learn quick tricks to check if numbers divide evenly without doing long division.

Explore the lesson 4 min read
SSAT Middle Level illustration with a test notebook, quantitative function, and reading book
Inside this lesson 10 sections 5 questions 6 figures

SECTION 1

A Quick Look at the Past

Long ago, people needed fast ways to check big numbers. Divisibility rules are like secret shortcuts from ancient math whizzes. They help you see if one number goes into another without leftovers.

300 BC • Euclid's Ideas

Greek mathematician Euclid wrote about factors in his book. He helped start number rules.

800 AD • Indian Math Wizards

Scholars in India shared digit sum tricks for 3 and 9. These spread worldwide.

1600s • Europe Adopts Them

Math books in Europe added rules for 2, 5, and 10. Kids like you started learning them.

Today • SSAT Prep

These rules save time on tests like the SSAT. You can master them now!

Before computers, these rules helped traders and builders. Now they help you on tests. Let's dive in! You will feel like a math detective.

SECTION 2

What Are Divisibility Rules?

A divisibility rule tells if one number divides evenly into another. It means no remainder, like 12 into 36 perfectly.

1

Even Split

The big number splits into equal groups with nothing left over.
2

Fast Check

Skip long division. Use digit tricks instead. It's like a game cheat code.
3

Common Ones

Rules work best for small numbers like 2, 3, 5, and 9.
4

Practice Wins

Memorize them like your favorite video game moves. You will ace the SSAT!
⚡

Quick Tip

Divisibility rules are like checking if a backpack fits in a locker. Peek at key spots instead of trying every time.

SECTION 3

See the Rules in Action

Divisibility Rules — See the Rules in Action Quick tests to check if a number divides evenly — no long division needed! Example Number: 2,520 (divisible by all rules below!) 2 Divide by 2 Rule: Last digit is even (0, 2, 4, 6, or 8) 2,520 → last digit is 0 0 is even ✓ 3 Divide by 3 Rule: Sum of digits is divisible by 3 2+5+2+0 = 9 9 ÷ 3 = 3 ✓ 4 Divide by 4 Rule: Last 2 digits divisible by 4 Last 2 digits: 20 20 ÷ 4 = 5 ✓ 5 Divide by 5 Rule: Last digit is 0 or 5 2,520 → ends in 0 0 is 0 or 5 ✓ 6 Divide by 6 Rule: Divisible by both 2 AND 3 By 2? ✓ By 3? ✓ Both pass! ✓ 8 Divide by 8 Rule: Last 3 digits divisible by 8 Last 3 digits: 520 520 ÷ 8 = 65 ✓ 9 Divide by 9 Rule: Sum of digits is divisible by 9 2+5+2+0 = 9 9 ÷ 9 = 1 ✓ 10 Divide by 10 Rule: Last digit is 0 2,520 → ends in 0 Easiest rule! ✓ Summary for 2,520: Divisible by 2 3 4 5 6 8 9 10 — that's 8 out of 8 rules! 💡 Pro Tips for Quick Checks Rule of 6 = Rule of 2 + Rule of 3 combined. If both pass, it's divisible by 6! Rules of 3 and 9 are similar — just check if the digit sum divides by 3 or 9. Rule of 4 checks last two digits; Rule of 8 checks last three digits — look for the pattern!
This diagram shows examples for rules 2, 3, 5, and 9. Highlights mark the key digits to check.

Look at the diagram above. Each box shows a number and the trick to check it. Practice spotting the clues! It's fun like finding hidden objects in a picture.

SECTION 4

How the Math Works

Behind the tricks is simple math called modulo (short for remainder). If n ÷ d has no remainder, it's divisible.

DIVISIBILITY TEST

nmodd=0n mod d = 0

n = big number, d = divisor like 3. Zero remainder means yes!

SUM FOR 3

Sum(digitsofn)mod3=0Sum(digits of n) mod 3 = 0

Why? 10 ≡ 1 mod 3, so powers of 10 act like 1. Easy add-up!

These rules use patterns in base 10. You don't need to know why yet. Just use them to save time! Great job learning this.

SECTION 5

All the Main Rules

Divisibility Rules — All the Main Rules Quick tests to check if a number divides evenly — no calculator needed! Divisor The Rule (How to Test) Example Result 2 Last digit is even (ends in 0, 2, 4, 6, or 8) 348 → last digit 8 ✓ Yes 3 Sum of digits is divisible by 3 (add all digits, check if ÷3) 129 → 1+2+9 = 12 ✓ Yes 4 Last two digits divisible by 4 (check only the final two digits) 1,524 → 24 ÷ 4 = 6 ✓ Yes 5 Last digit is 0 or 5 (just look at the ones place) 735 → last digit 5 ✓ Yes 6 Divisible by both 2 AND 3 (pass both the 2-rule and 3-rule) 126 → even, 1+2+6=9 ✓ Yes 7 Double last digit, subtract from rest (if result is ÷7, the number is too) 203 → 20−6 = 14 ✓ Yes 8 Last three digits divisible by 8 (check only the final three digits) 1,160 → 160÷8 = 20 ✓ Yes 9 Sum of digits is divisible by 9 (like the 3-rule, but check for ÷9) 162 → 1+6+2 = 9 ✓ Yes 10 Last digit is 0 (the easiest rule of all!) 4,530 → last digit 0 ✓ Yes 💡 Tip: The rules for 3 and 9 are similar — just change what the digit sum must divide by!
Chart lists rules for 2,3,4,5,6,9,10 with tests and examples. Each row highlights one rule.

The chart above breaks down each rule. Pick the one you need and check fast. You are getting good at this! Keep going.

SECTION 6

Step-by-Step: Test 456 for 3, 4, 9

Is 456 divisible by 3? By 4? By 9?

  1. 1

    Step 1: For 3

    Sum digits: 4 + 5 + 6 = 15. 15 ÷ 3 = 5, no remainder. Yes!
    Yes for 3
  2. 2

    Step 2: For 4

    Last two digits: 56. 56 ÷ 4 = 14, exact. Yes!
    Yes for 4
  3. 3

    Step 3: For 9

    Sum is 15. 15 ÷ 9 = 1 remainder 6. No.
    No for 9

See how easy? You just used three rules. You can do this on any number!

SECTION 7

Why Rules Rock (and Limits)

StrengthLimit
Super fast for big numbers. No rule for 7 or 11 easily.
Saves time on SSAT tests. Need to memorize a few.
Fun and like magic tricks. Always double-check with division sometimes.

Pros and cons of using rules.

🏆

Key takeaway

Rules are your speedy sidekick in math battles, like a quick dodge in soccer.

SECTION 8

Next Level: Bigger Ideas

Basic RuleAdvanced Way
Sum digits for 3/9Prime factors (3×3=9)
Last digit for 2/5Modulo math for any number
Quick testFull prime check

Rules lead to cool stuff like finding primes. You are on your way there! Keep practicing.

SECTION 9

Test Yourself!

Question 1: CONCEPTUAL

Which rule checks if a number is divisible by 5? (A) Sum of digits ÷5 (B) Ends in 0 or 5 (C) Last digit even (D) Last two digits ÷5 (E) Ends in even digit
Show answer
B is correct. A number is divisible by 5 if it ends in 0 or 5. A is wrong (that's not a rule). C is for 2. D mixes with 4. E confuses with 2 or 10. Great start!

Question 2: BASIC CALCULATION

Is 234 divisible by 2? (A) Yes (B) No (C) Check sum (D) Check last two (E) Ends in 5
Show answer
A is correct. Last digit 4 is even, so yes for 2. B wrong. C for 3. D for 4. E for 5. Easy one—you got it!

Question 3: INTERMEDIATE

Is 567 divisible by 3? Sum=5+6+7=18. (A) Yes (B) No (C) Check for 9 (D) Last digit (E) No, 18 not ÷3
Show answer
A is correct. 18÷3=6 exactly. B wrong. C extra step. D wrong rule. E silly—18 is ÷3. Nice sum work!

Question 4: APPLIED

Which is divisible by 4? (A) 123 (B) 456 (C) 789 (D) 101 (E) 222
Show answer
B correct. 56÷4=14. A 23÷4=5.75 no. C 89÷4=22.25 no. D 01÷4=0.25 no. E 22÷4=5.5 no. Smart check!

Question 5: CRITICAL THINKING

Is 999 divisible by 9 and 3? (A) 9 yes, 3 no (B) Both yes (C) Both no (D) 9 no, 3 yes (E) Check 2
Show answer
B correct. Sum=27, 27÷9=3 and ÷3=9. A/D wrong since 9 implies 3. C false. E wrong rule. You nailed the connection!
💡

You Rock!

Review wrong ones. Practice makes perfect for SSAT.

SUMMARY

Review Time

Summary

Master divisibility rules for 2 (even end), 3/9 (digit sum), 4 (last two), 5/10 (0/5 end), 6 (2+3). They save test time!

Use visuals and steps. You are ready for SSAT success. Believe in yourself!

  • ✓ Practice daily
  • ✓ Check big numbers fast
  • ✓ Ace the Quantitative section

Varsity Tutors • SSAT Middle Level • Use divisibility rules to test a number.