Use divisibility and factors to reason about numbers. |...

SECTION 1

Where Did Divisibility Come From?

People have been breaking numbers apart for thousands of years! Long ago, ancient mathematicians noticed that some numbers can be split into equal groups and some cannot. This idea is called divisibility — it means one number can be divided by another with nothing left over.

Imagine you have 12 cookies. You can share them equally among 2 friends, 3 friends, 4 friends, or 6 friends. But 12 cookies cannot be shared equally among 5 friends — someone would get fewer! That's divisibility in action.

300 BC

Euclid's Elements

A Greek mathematician named Euclid wrote one of the first math books. He showed how to find factors of numbers.

200 BC

Sieve of Eratosthenes

Eratosthenes invented a clever method to find prime numbers — numbers that have only two factors: 1 and themselves.

500 AD

Indian Mathematicians

Mathematicians in India created divisibility rules — quick tricks to check if a number divides evenly.

Today

You on the ISEE!

These same ideas show up on the ISEE test. Knowing divisibility tricks helps you solve problems faster!

So here's the big question: how can you quickly tell if one number divides evenly into another? Let's find out!

SECTION 2

Core Ideas: Factors, Divisibility & Multiples

Before we start solving problems, let's learn three important words. These are the building blocks for everything in this lesson.

Factor

A factor is a number that divides evenly into another number. For example, 3 is a factor of 12 because 12 ÷ 3 = 4 with no remainder.

Divisible

A number is divisible by another if there is no remainder. 20 is divisible by 5 because 20 ÷ 5 = 4 exactly.

Multiple

A multiple is the result of multiplying a number by a whole number. 15 is a multiple of 5 because 5 × 3 = 15.

Factor Pair

A factor pair is two numbers that multiply together to make a number. 3 × 4 = 12, so (3, 4) is a factor pair of 12.

✦ KEY TAKEAWAY

Think of factors like puzzle pieces. Every number can be "snapped apart" into factor pairs — just like a LEGO tower can be broken into smaller stacks. If you know the factor pairs, you can quickly tell if one number divides into another!

💡 ISEE Test Tip

On the ISEE, you won't see the word "divisibility" in every problem. Instead, look for clues like "divides evenly," "no remainder," "shared equally," or "is a factor of." These all mean the same thing!

SECTION 3

See It! The Factor Rainbow

One of the best ways to find all the factors of a number is to draw a factor rainbow. You list numbers from small to large, then connect the factor pairs with arcs — like a rainbow! Let's see the factor rainbow for the number 24.

Each colorful arc connects a factor pair that multiplies to make 24. The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, and 24. That's 8 factors!

Notice how each arc connects two numbers that multiply to 24. Start with 1 on the left and work your way in. When the arcs start overlapping, you've found all the factor pairs!

SECTION 4

Divisibility Rules — Your Secret Shortcuts

You don't always need to do long division to check divisibility. There are quick rules that save you tons of time on the ISEE. Let's learn the most important ones!

DIVISIBLE BY 2

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

These are called even numbers. Example: 346 is divisible by 2 because the last digit (6) is even.

DIVISIBLE BY 3

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

Example: Is 243 divisible by 3? Add the digits: 2 + 4 + 3 = 9. Since 9 is divisible by 3, yes!

DIVISIBLE BY 5

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

Example: 475 ends in 5, so it is divisible by 5. But 472 ends in 2, so it is NOT divisible by 5.

DIVISIBLE BY 10

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

Example: 530 ends in 0, so it is divisible by 10. Easy!

⚡ ISEE Speed Trick

On the ISEE, if a problem asks "Which number is divisible by 6?" remember that 6 = 2 × 3. So the number must pass BOTH the rule for 2 AND the rule for 3. Check the last digit first (rule for 2), then add up the digits (rule for 3). This saves time!

SECTION 5

Divisibility Rules at a Glance

Here is a handy chart that shows all the divisibility rules you need for the ISEE. You can also see which rules work together!

Keep this chart in your mind! The rules for 2, 5, and 10 only need the last digit. The rules for 3 and 9 need you to add the digits. The rule for 6 uses two rules together.

Practice checking divisibility with these examples!

Number	Divisible by 2?	Divisible by 3?	Divisible by 5?
45	No (ends in 5)	Yes (4+5=9)	Yes (ends in 5)
72	Yes (ends in 2)	Yes (7+2=9)	No (ends in 2)
130	Yes (ends in 0)	No (1+3+0=4)	Yes (ends in 0)

SECTION 6

Step-by-Step: Finding All Factors

Let's solve a problem together, just like you would on the ISEE. Take your time and follow each step.

📝 Sample Problem

Maria wants to arrange 36 stickers into equal rows. Which of the following is NOT a way she can arrange them? (A) 4 rows of 9 (B) 6 rows of 6 (C) 5 rows of 7 (D) 3 rows of 12

Solving the Sticker Problem

Step 1 — Understand the ProblemMaria has 36 stickers. She wants equal rows with no stickers left over. We need to find which answer does NOT divide evenly into 36.

Step 2 — Check Each AnswerChoice A: 4 × 9 = 36 ✓ That works! Choice B: 6 × 6 = 36 ✓ That works! Choice C: 5 × 7 = 35, NOT 36 ✗ That does NOT work! Choice D: 3 × 12 = 36 ✓ That works!

Step 3 — Confirm Your Answer5 × 7 = 35, which is one less than 36. So 5 and 7 are NOT factors of 36. Maria would have 1 sticker left over.

The answer is (C) 5 rows of 7.

🎯 TEST STRATEGY

When a problem says "NOT," check every answer choice. Multiply the two numbers in each choice. If the product does not match the total, that's your answer. Circle the word NOT so you don't forget what you're looking for!

SECTION 7

Strengths & Limits of Each Strategy

You now have several tools for working with factors and divisibility. Let's see when each one works best!

Pick the fastest strategy for each problem!

Strategy	When It Helps	Watch Out For
Divisibility Rules	Quick checks with large numbers — no long division needed!	Only works for 2, 3, 4, 5, 6, 9, and 10. For other numbers, you need to divide.
Factor Pairs	Finding ALL factors of a number. Great for problems that say "how many ways."	Don't forget 1 and the number itself! They are always factors.
Skip Counting	Listing multiples. Count by 3s, 4s, 5s to find multiples of a number.	Can take a while for large numbers. Use divisibility rules as a shortcut.
Multiply & Check	When answer choices are given. Multiply each choice and see if it matches.	Read the question carefully — are they asking IS or IS NOT a factor?

🧠 REMEMBER

On the ISEE, there's no penalty for guessing! If you're stuck, use process of elimination. Cross out answer choices you know are wrong, then pick from what's left. Even eliminating one choice improves your odds!

SECTION 8

Connecting to Bigger Ideas

The skills you're learning now connect to important ideas you'll use in later grades. Here's a sneak peek at how factors and divisibility grow up!

Today's lesson is the foundation for all of this!

What You Know Now	What Comes Next
Finding factors of a number	Greatest Common Factor (GCF) — the biggest factor two numbers share
Listing multiples	Least Common Multiple (LCM) — the smallest multiple two numbers share
Divisibility rules	Simplifying fractions — divide the top and bottom by a common factor
Numbers with only 2 factors (1 and itself)	Prime numbers and prime factorization — breaking any number into prime pieces

For now, just focus on finding factors, using divisibility rules, and checking if numbers divide evenly. You're building a strong foundation for all the math ahead!

SECTION 9

Practice Problems

Time to show what you know! Try these five problems. They get harder as you go. Remember — on the ISEE, always answer every question, even if you have to guess!

Which of the following numbers is a factor of 18?

Which number is divisible by both 2 and 5?

How many factors does the number 20 have?

A teacher has 42 pencils to share equally among students. She does NOT want any pencils left over. Which of the following is NOT a possible number of students?

Sam is thinking of a number between 30 and 50. His number is divisible by 3 and by 4, is NOT divisible by 5, and is less than 40. What is Sam's number?

SUMMARY

Let's Review!

In this lesson, you learned that a factor is a number that divides evenly into another number. You discovered how to use divisibility rules as quick shortcuts: check the last digit for 2, 5, and 10, and add the digits for 3 and 9. You practiced finding all factor pairs using the factor rainbow, and you learned that multiples are the flip side of factors.

For the ISEE, remember these key strategies: use process of elimination to cross out wrong answers, multiply and check when answer choices are given, and always answer every question — there's no penalty for guessing! You've got this!