Identify factors and multiples of whole numbers.

SECTION 1

Historical Context & Motivation

Long ago, people needed to share things equally, like crops or land. Factors helped them divide items without leftovers. This idea started in ancient times.

3000 BC

Ancient Egypt

Egyptians used factors to build pyramids with equal stones.

300 BC

Euclid's Work

Greek mathematician Euclid wrote about divisors in his book Elements.

1202

Fibonacci

Italian scholar shared factoring methods from Arab math.

Today

SSAT Prep

You use factors and multiples to ace test questions!

These tools solve real problems like splitting teams in sports. Now you can use them on the SSAT!

SECTION 2

Core Principles & Definitions

A factor of a whole number is any number that divides it evenly with no remainder. Like 2 and 3 are factors of 6.

Factors

Numbers that multiply to make the original. Test by dividing.

Multiples

Original number times any whole number. Like 6, 12, 18 for 6.

Key Rule

Every number is a factor and multiple of itself.

1 and Itself

1 and the number are always factors.

⚽ Quick Analogy

Factors are like ways to split a soccer team into equal groups. Multiples are scores like 2-0, 4-0 from kicking twice.

SECTION 3

Visual Explanation

Rectangles show factor pairs of 12. Each has the same area but different lengths and widths.

See how factors pair up to fill the same space? This visual helps you spot them quickly on tests. You got this!

SECTION 4

Mathematical Framework

To find factors, divide the number by whole numbers from 1 up. Stop at its square root for speed.

FACTOR TEST

If n ÷ d = whole number (no remainder), then d is a factor of n

n = target number, d = divisor to test

MULTIPLE FORMULA

Multiple = n × k (where k is any whole number ≥ 1)

Examples: multiples of 5 are 5×1=5, 5×2=10, etc.

Practice dividing step by step. It builds your math muscles for SSAT success!

SECTION 5

Detailed Breakdown

Dots mark multiples of 4. The dashed line shows the skip pattern.

Multiples follow a jumping pattern on the line. Factors pair from both ends. Spot these patterns to win!

SECTION 6

Worked Example

List all factors of 18

Step 1: List divisors from 1Try 1: 18 ÷ 1 = 18 (yes). 2: 18 ÷ 2 = 9 (yes).

1, 2

Step 2: Continue to square root (≈4.24)3: 18 ÷ 3 = 6 (yes). 4: 18 ÷ 4 = 4.5 (no).

1, 2, 3

Step 3: Pair and list allPairs: 1&18, 2&9, 3&6. Factors: 1,2,3,6,9,18.

1, 2, 3, 6, 9, 18

See the pattern? Practice this method and you'll crush SSAT questions.

SECTION 7

Strengths & Common Errors

Avoid pitfalls to score higher

Strength	Common Mistake	Fix
Quick to check with division	Forgetting 1 and itself	Always include them!
Visual patterns easy to spot	Listing beyond square root	Stop at √n and pair up

✦ KEY TAKEAWAY

Like video game levels, master basics before advancing. You're building pro skills!

SECTION 8

Connection to Advanced Topics

Basic	Advanced
List all factors	Prime factorization
Find some multiples	GCF and LCM

These basics lead to greatest common factor (GCF) for simplifying fractions. Keep going—you're ready!

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

Which is NOT a factor of 10? A) 1 B) 2 C) 5 D) 10 E) 15

PROBLEM 2 — BASIC CALCULATION

What is the smallest multiple of 7 greater than 20? A) 21 B) 24 C) 28 D) 35 E) 42

PROBLEM 3 — INTERMEDIATE

How many factors does 12 have? A) 3 B) 4 C) 5 D) 6 E) 7

PROBLEM 4 — APPLIED

A recipe needs groups of 9 cookies per tray. Which tray size fits 36 cookies exactly? A) 3 B) 4 C) 6 D) 9 E) 12

PROBLEM 5 — CRITICAL THINKING

Which number has the most factors between 1 and 20? A) 12 B) 14 C) 15 D) 16 E) 18

SUMMARY

Lesson Summary

You can now identify factors by dividing evenly and multiples by skip-counting. Use visuals and pairs for speed.

Practice daily like training for a game. You're SSAT ready—believe it!