Apply divisibility and factor reasoning. - ISEE Middle Level Quantitative Reasoning

SECTION 1

Where Did Divisibility Rules Come From?

People have been breaking numbers apart for thousands of years. Ancient traders needed quick ways to split goods into equal groups. Imagine dividing 60 loaves of bread among workers — you'd want to know instantly whether the split comes out even!

Over time, mathematicians noticed patterns that made this easier. They created divisibility rules — simple tests you can do in your head. These rules save tons of time, especially on a test like the ISEE where you can't use a calculator.

~300 BC • Euclid's Elements

The Greek mathematician Euclid wrote down the first formal rules about factors and prime numbers. His method for finding the greatest common factor is still used today.

~200 BC • Sieve of Eratosthenes

Eratosthenes invented a clever way to filter out all prime numbers from a list. This "sieve" uses divisibility to cross off multiples one by one.

500s AD • Hindu-Arabic Number System

Indian mathematicians developed the place-value system (ones, tens, hundreds) that makes our modern divisibility rules possible. The rule for dividing by 5, for example, depends on the ones digit.

1200s • Fibonacci Spreads the System

Italian mathematician Fibonacci introduced the Hindu-Arabic numerals to Europe. Merchants quickly adopted divisibility shortcuts for trade calculations.

Today, divisibility and factor reasoning appear on almost every standardized math test. On the ISEE, these ideas show up in questions about remainders, greatest common factors, least common multiples, and prime factorization. Let's build the skills you need to handle all of them.

SECTION 2

Core Principles of Divisibility

Before we jump into tricks and shortcuts, let's nail down the key vocabulary. These are the building blocks you'll use throughout the lesson.

Divisible

A number is divisible by another if the division has no remainder. For example, 18 is divisible by 3 because 18 ÷ 3 = 6 exactly.

Factor

A factor of a number divides into it evenly. The factors of 12 are 1, 2, 3, 4, 6, and 12. Factors always come in pairs: 2 × 6 = 12.

Prime Number

A prime number has exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13. The number 1 is NOT prime.

Prime Factorization

Breaking a number into a product of primes. For example, 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5. Every whole number greater than 1 has exactly one prime factorization.

GCF and LCM

The Greatest Common Factor (GCF) is the largest factor two numbers share. The Least Common Multiple (LCM) is the smallest number both divide into evenly.

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KEY TAKEAWAY

Think of factors like ingredients in a recipe. If 12 cookies need 2 cups of flour, 3 eggs, and 4 tablespoons of butter, each of those amounts (2, 3, 4) divides evenly into 12. They're all factors of 12! On the ISEE, spotting factors quickly lets you simplify fractions, compare quantities, and solve problems faster.

SECTION 3

See the Divisibility Rules in Action

The diagram below shows the most important divisibility rules for the ISEE. Each rule gives you a quick mental test. Memorize these — they'll save you precious seconds on test day.

Each card shows the divisor and the quick test to check. For ÷ 6, you combine the tests for 2 and 3. The example at the bottom shows how to apply two rules together.

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ISEE TEST TIP

On the ISEE, you won't see a question that just says "Is 372 divisible by 6?" Instead, divisibility shows up hidden inside word problems and quantitative comparisons. Knowing these rules by heart lets you work faster without a calculator.

SECTION 4

The Math Behind Factors and Multiples

Now let's look at the formulas and methods you'll actually use on ISEE problems. The key tools are prime factorization, GCF, and LCM.

PRIME FACTORIZATION

60 = 2² × 3 × 5

Break a number into a product of prime numbers. Start by dividing by the smallest prime (2), then try 3, 5, 7, and so on until you reach 1.

GREATEST COMMON FACTOR (GCF)

GCF = product of shared prime factors (use lowest power)

Example: 36 = 2² × 3² and 48 = 2⁴ × 3. Shared primes: 2 and 3. Take the lower power of each: 2² × 3¹ = 4 × 3 = 12. So GCF(36, 48) = 12.

LEAST COMMON MULTIPLE (LCM)

LCM = product of all prime factors (use highest power)

Example: 36 = 2² × 3² and 48 = 2⁴ × 3. Take the higher power of each prime: 2⁴ × 3² = 16 × 9 = 144. So LCM(36, 48) = 144.

GCF × LCM SHORTCUT

GCF(a, b) × LCM(a, b) = a × b

Check: GCF(36, 48) × LCM(36, 48) = 12 × 144 = 1,728. And 36 × 48 = 1,728. ✓ This shortcut can save time if you already know one value and need the other.

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REMEMBER

GCF uses the LOWEST powers (the factors they share). LCM uses the HIGHEST powers (everything you need to build both numbers). A helpful memory trick: GCF = "Go for the Common, Few" and LCM = "Lots — Choose the Max."

SECTION 5

Factor Trees and Finding All Factors

A factor tree is a visual way to break a number into its prime factors. You keep splitting until every "leaf" of the tree is a prime number. The diagram below shows how to build a factor tree for 72.

Start at the top with 72. Split it into 8 × 9. Keep splitting each composite number until you reach primes (circled at the leaves). Collect all the primes: three 2's and two 3's.

How to Count All Factors of a Number

Once you have the prime factorization, there's a neat trick to count how many factors a number has. Add 1 to each exponent and multiply those results together.

COUNTING FACTORS

72 = 2³ × 3² → (3 + 1) × (2 + 1) = 4 × 3 = 12 factors

The 12 factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. On the ISEE, a question might ask "How many factors does 72 have?" This formula gets you there fast.

Number	Prime Factorization	Number of Factors
24	2³ × 3	(3+1)(1+1) = 8
36	2² × 3²	(2+1)(2+1) = 9
50	2 × 5²	(1+1)(2+1) = 6
100	2² × 5²	(2+1)(2+1) = 9

Common numbers and their factor counts

SECTION 6

Worked Example: ISEE-Style Problem

Let's walk through a realistic ISEE problem step by step. Pay attention to how we use divisibility rules and factor reasoning together.

Finding the GCF to Simplify a Fraction

1
Step 1 — Read the Problem

A teacher has 84 pencils and 60 erasers. She wants to make identical supply bags with no supplies left over. What is the greatest number of bags she can make?
2
Step 2 — Identify the Math

We need the greatest number that divides evenly into both 84 and 60. That's the Greatest Common Factor (GCF) of 84 and 60.
3
Step 3 — Prime Factorize Each Number

84 = 2 × 42 = 2 × 2 × 21 = 2 × 2 × 3 × 7 = 2² × 3 × 7. And 60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5 = 2² × 3 × 5.

84 = 2² × 3 × 7 | 60 = 2² × 3 × 5
4
Step 4 — Find Shared Primes with Lowest Powers

Both share the prime 2 (lowest power: 2²) and the prime 3 (lowest power: 3¹). The 7 in 84 and the 5 in 60 are NOT shared, so we skip them.
5
Step 5 — Multiply Shared Factors

GCF = 2² × 3 = 4 × 3 = 12.

The teacher can make 12 bags.
6
Step 6 — Check

84 ÷ 12 = 7 pencils per bag. 60 ÷ 12 = 5 erasers per bag. Both divide evenly with nothing left over. ✓

SECTION 7

ISEE Strategies: When to Use Which Method

On the ISEE, time is tight. Knowing which method to use is just as important as knowing how to do the math. Here's a comparison of your main tools.

Method	Best For	Watch Out
Divisibility Rules	Quick checks: "Is this divisible by 3?" Eliminating wrong answer choices.	Only works for specific divisors (2, 3, 4, 5, 6, 8, 9, 10). No rule for 7.
Listing Factors	Small numbers (under 50). Finding GCF when you need to see all factors.	Slow for big numbers. Easy to miss a factor pair.
Prime Factorization	GCF and LCM of larger numbers. Counting total factors.	Takes more steps. Don't forget to check all small primes.
Testing Answer Choices	When you're stuck! Plug each answer in and see which one works.	Uses more time. Start with the middle answer choice to eliminate faster.

Choosing the right tool for each ISEE question

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ISEE STRATEGY TIP

Think of your strategies like apps on your phone. You don't open the camera app to send a text! Divisibility rules are your "quick check" app — use them first to eliminate wrong answers. Prime factorization is your "deep analysis" app — use it when you need exact GCF or LCM values. And remember: there's no penalty for guessing on the ISEE, so always pick an answer even if you're unsure.

SECTION 8

Connecting Factors to Other ISEE Topics

Factor reasoning doesn't live in isolation. It connects to many other topics you'll see on the ISEE. Understanding these connections makes you a more flexible problem solver.

Factor Skill	Connected ISEE Topic	How They Connect
GCF	Simplifying fractions	Divide top and bottom by the GCF to reduce a fraction to lowest terms.
LCM	Adding/subtracting fractions	The LCM of the denominators gives you the least common denominator.
Divisibility	Remainders	If a number is NOT divisible, the leftover is the remainder.
Prime factorization	Exponents and powers	Writing 72 = 2³ × 3² uses exponent notation you also need for algebra.
Factor pairs	Area and perimeter	If a rectangle has area 36, its length and width must be a factor pair of 36.

Factor reasoning connects to many ISEE topics

As you move into higher-level math, factors become even more important. In algebra, you'll factor expressions like x² + 5x + 6 into (x + 2)(x + 3). The thinking is the same — you're looking for numbers that multiply together to give a result. Building strong factor skills now sets you up for success later.

SECTION 9

Practice Problems

Now it's your turn! Try these five problems. They start easier and get harder, just like the ISEE. Remember: there's no penalty for guessing, so always choose an answer. Use process of elimination to cross out wrong choices first.

Question 1: CONCEPTUAL

Which of the following numbers is divisible by both 3 and 4? (A) 16 (B) 21 (C) 36 (D) 44

A. 16
B. 21
C. 36
D. 44

Show answer (C)

The correct answer is (C) 36. To be divisible by both 3 and 4, a number must pass both tests. For ÷ 3, add the digits and check if the sum is divisible by 3. For ÷ 4, check if the last two digits form a multiple of 4. Let's test each: (A) 16: digits sum to 7, which is NOT divisible by 3. Eliminated. (B) 21: digits sum to 3 ✓, but 21 ÷ 4 = 5 remainder 1. Not divisible by 4. Eliminated. (C) 36: digits sum to 9, and 9 ÷ 3 = 3 ✓. Also 36 ÷ 4 = 9 ✓. Passes both! (D) 44: digits sum to 8, and 8 ÷ 3 = 2 remainder 2. Not divisible by 3. Eliminated.

Question 2: BASIC CALCULATION

What is the greatest common factor (GCF) of 28 and 42? (A) 2 (B) 7 (C) 14 (D) 28

A. 2
B. 7
C. 14
D. 28

Show answer (C)

The correct answer is (C) 14. First, find the prime factorizations: 28 = 2² × 7 and 42 = 2 × 3 × 7. The shared primes are 2 (lowest power: 2¹) and 7 (lowest power: 7¹). So GCF = 2 × 7 = 14. (A) 2 is a common factor, but not the greatest one. (B) 7 is a common factor, but not the greatest one. (D) 28 does not divide evenly into 42 (42 ÷ 28 = 1 remainder 14), so 28 cannot be a common factor.

Question 3: INTERMEDIATE

Quantitative Comparison: Column A: The number of factors of 48 Column B: The number of factors of 50 (A) Column A is greater. (B) Column B is greater. (C) The two quantities are equal. (D) Cannot be determined.

A. Column A is greater.
B. Column B is greater.
C. The two quantities are equal.
D. Cannot be determined.

Show answer (A)

The correct answer is (A). Column A: 48 = 2⁴ × 3. Number of factors = (4+1)(1+1) = 5 × 2 = 10. Column B: 50 = 2 × 5². Number of factors = (1+1)(2+1) = 2 × 3 = 6. Since 10 > 6, Column A is greater.

Question 4: APPLIED

Two buses leave the station at 8:00 AM. Bus A returns every 12 minutes, and Bus B returns every 18 minutes. At what time will both buses be back at the station at the same time for the first time after 8:00 AM? (A) 8:18 AM (B) 8:30 AM (C) 8:36 AM (D) 8:54 AM

A. 8:18 AM
B. 8:30 AM
C. 8:36 AM
D. 8:54 AM

Show answer (C)

The correct answer is (C) 8:36 AM. We need the Least Common Multiple (LCM) of 12 and 18. Prime factorizations: 12 = 2² × 3 and 18 = 2 × 3². Take the highest power of each prime: 2² × 3² = 4 × 9 = 36. So both buses return together after 36 minutes, which is 8:36 AM. (A) 8:18 AM is only 18 minutes — Bus B returns, but Bus A does not (12 doesn't divide into 18 evenly). (B) 8:30 AM is 30 minutes — neither 12 nor 18 divides evenly into 30. (D) 8:54 AM is 54 minutes — while both buses do return at this time (it's a common multiple), it's not the FIRST time they meet.

Question 5: CRITICAL THINKING

Quantitative Comparison: n is a positive integer greater than 1. Column A: The number of factors of n² Column B: The number of factors of 2n (A) Column A is greater. (B) Column B is greater. (C) The two quantities are equal. (D) Cannot be determined.

A. Column A is greater.
B. Column B is greater.
C. The two quantities are equal.
D. Cannot be determined.

Show answer (D)

The correct answer is (D). The relationship depends on the value of n. Test n = 2: Column A — n² = 4, factors of 4 are 1, 2, 4 → 3 factors. Column B — 2n = 4, factors of 4 are 1, 2, 4 → 3 factors. The quantities are equal. Test n = 3: Column A — n² = 9, factors of 9 are 1, 3, 9 → 3 factors. Column B — 2n = 6, factors of 6 are 1, 2, 3, 6 → 4 factors. Column B is greater. Since different values of n give different results, the answer is (D). This is a classic ISEE quantitative comparison strategy: when variables are involved, always test at least two different values before choosing your answer.

SUMMARY

Pulling It All Together

Summary

You've learned the essential tools for tackling ISEE questions about factors and divisibility. Divisibility rules let you quickly check whether a number divides evenly by 2, 3, 4, 5, 6, 8, 9, or 10 — just by looking at the digits. Prime factorization breaks any number into its prime building blocks, which you can use to find the GCF (shared primes, lowest powers) and LCM (all primes, highest powers). You also learned the factor counting trick: add 1 to each exponent and multiply.

On the ISEE, these skills show up in word problems about equal groups, simplifying fractions, and finding common denominators. For quantitative comparisons, remember to test multiple values when variables are involved — the answer might be (D). Always use process of elimination to cross out wrong answers, and never leave a question blank. You've got this!