Determine whether a number is prime or composite.

SECTION 1

Historical Context & Motivation

Long ago, people noticed some numbers are special. They called them prime numbers because no other numbers can divide them evenly except 1 and themselves. This idea helps us understand all whole numbers better. Ancient Greeks like Euclid studied primes over 2,000 years ago.

300 BCE

Euclid's Proof

Euclid proved there are infinitely many primes. He showed primes never run out.

240 BCE

Eratosthenes' Sieve

Eratosthenes created a method to find primes. His sieve knocks out multiples fast.

2000 CE

Modern Use

Computers find huge primes for secure codes. Primes protect your online games.

Why care on the SSAT? Spotting primes quickly builds your math speed. You will see them in questions about factors and multiples.

SECTION 2

Core Principles & Definitions

A prime number is a whole number greater than 1. It has exactly two different factors: 1 and itself. Like 2, 3, 5, or 7.

A composite number has more than two factors. Think of 4 (1, 2, 4), 6 (1, 2, 3, 6), or 9 (1, 3, 9). Number 1 is neither prime nor composite.

Prime

Only two factors. Like a solo superhero.

Composite

Three or more factors. Like a team sport.

Neither

1 has just one factor. Special case!

Test Rule

Check divisors up to square root.

⚽ Quick Analogy

Primes are like a soccer player who can't pass the ball to anyone but themselves or the goalie. Composites pass to lots of teammates!

SECTION 3

Visual Explanation

The diagram shows factor trees for 13 (prime: only two ends) and 12 (composite: many ends). Arrows point to factors.

See how the prime tree stops quick with just two branches? Composites keep splitting like a busy family tree. This visual helps you spot them fast.

SECTION 4

Mathematical Framework

To test if n > 1 is prime, check if any number from 2 to √n divides it evenly. √n means square root of n. No divisors? It's prime!

PRIME TEST

If n has no divisors d where 2 ≤ d ≤ √n, then n is prime.

Example: For 17, √17 ≈ 4.1. Check 2,3,4. None divide 17.

Even numbers >2 are composite (div by 2).
Ends in 5? Div by 5 (composite).
Sum digits div by 3? Composite.

SECTION 5

Detailed Breakdown & Classification

Sieve method: Keep primes in green, cross out their multiples to reveal composites.

The sieve is like a game of tag. Primes tag their multiples out. What's left standing are primes!

SECTION 6

Worked Example

Is 37 Prime?

Step 1: Check if >137 > 1. Good start.

Yes

Step 2: √37 ≈ 6.1Test divisors 2 to 6.

Step 3: Test 2,3,537 ÷ 2 = 18.5 (no). 37 ÷ 3 ≈12.333 (no). 37 ÷ 5 =7.4 (no).

No divisors!

Conclusion37 is prime. Great job practicing!

SECTION 7

Prime vs Composite Properties

Compare to see patterns fast.

Property	Prime	Composite
Factors	Exactly 2	More than 2
Examples	2,3,5,11	4,6,9,15
Multiples	Build all numbers	Made from primes

✦ KEY TAKEAWAY

Primes are the atoms of numbers. Composites are molecules made from them. You're now a number detective!

SECTION 8

Connection to Advanced Theory

Basic	Advanced
Spot prime or composite	Prime factorization (break into primes)
Test up to √n	Huge primes for computer security (RSA)

Master this now, and factorization will feel easy later. Primes power codes that keep your games safe online.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

Which is true about prime numbers? A) They have no factors. B) They are greater than 1 with exactly two factors. C) They are even numbers. D) They include 1. E) They have three factors.

PROBLEM 2 — BASIC CALCULATION

Is 15 prime or composite? A) Prime B) Composite C) Neither D) Even E) Square

PROBLEM 3 — INTERMEDIATE

Is 29 prime? A) Yes B) No, divisible by 2 C) No, divisible by 3 D) No, divisible by 5 E) No, divisible by 7

PROBLEM 4 — APPLIED

A pizza box holds 25 slices. If not prime, what divides it besides 1 and 25? A) 2 B) 3 C) 4 D) 5 E) 6

PROBLEM 5 — CRITICAL THINKING

Which is composite? A) 31 B) 37 C) 43 D) 47 E) 91

SUMMARY

Lesson Summary

Master prime (>1, two factors) vs composite (more factors). Test divisors to √n or use sieve. You're ready for SSAT wins!

Practice spotting them like pros. Great job—you got this!