Determine whether numbers are prime or composite.

SECTION 1

Historical Context and Motivation

Understanding whether numbers are prime or composite dates back to ancient civilizations seeking patterns in nature and counting. Early mathematicians like the Greeks recognized primes as indivisible building blocks of all integers greater than 1. This distinction solved problems in divisibility and factorization essential for larger calculations. On the SSAT, mastering this helps you tackle quantitative reasoning efficiently.

300 BCE

Euclid's Elements

Euclid proves there are infinitely many primes, establishing their fundamental role in number theory.

240 BCE

Sieve of Eratosthenes

Eratosthenes invents a systematic method to identify primes up to any limit.

200 CE

Chinese Mathematics

Texts like the Suan shu shu describe prime testing for astronomical calculations.

1742

Goldbach Conjecture

Goldbach posits every even integer greater than 2 is sum of two primes, highlighting their importance.

These milestones reveal primes as the atoms of arithmetic, motivating efficient tests for SSAT problems. Recognizing composites speeds up elimination in multiple-choice questions. You can confidently approach any number on test day with these tools.

SECTION 2

Core Principles and Definitions

A prime number is a natural number greater than 1 with no positive divisors other than 1 and itself. In contrast, a composite number greater than 1 has additional divisors beyond 1 and itself. The number 1 is neither prime nor composite, serving as a unit rather than a building block. These definitions form the foundation for factorization and SSAT quantitative strategies.

Prime Definition

Divisors: only 1 and n. Example: 7 (checked up to √7 ≈ 2.6).

Composite Definition

Has divisors d where 1 < d < n. Example: 12 (2,3,4,6).

Testing Rule

Check divisibility by primes ≤ √n. Efficient for large numbers on SSAT.

Special Cases

1: neither. Even >2: composite. Ends in 5: divisible by 5.

✦ KEY TAKEAWAY

Think of primes like prime real estate— indivisible and valuable. Composites are like apartments, built from smaller prime units, just as 12 = 2 × 2 × 3.

SECTION 3

Visual Explanation with the Sieve

The sieve crosses multiples of each prime, leaving primes uncrossed. Notice 2 (blue), 3 (emerald), 5 (orange) as starting points.

This visual method, invented by Eratosthenes, efficiently identifies all primes up to a limit by eliminating composites. On the SSAT, you can mentally sieve small ranges for quick checks. Practice reveals patterns like even numbers beyond 2 always being composite. Build speed to handle test pressures confidently.

SECTION 4

Mathematical Framework

To determine if n > 1 is prime, verify no integer d satisfies 1 < d ≤ √n and n mod d = 0. This limits checks to primes up to √n, optimizing computation. Composites fail this test with at least one such d. SSAT problems often hide these in multi-step questions.

PRIME TEST

n is prime if ∀ d ∈ {2,3,...,⌊√n⌋}, n mod d ≠ 0

d: potential divisors; mod: remainder. Example for 17: check up to ⌊√17⌋=4 (2,3).

COMPOSITE FACTOR

n = p × q where 1 < p,q < n (p,q prime or composite)

Unique factorization theorem: every composite factors into primes.

Divisibility shortcuts: even (÷2), sum digits ÷3, ends 0/5 (÷5), etc.
Stop at √n; if no divisors, prime.
1 neither; 0/negatives not considered.

SECTION 5

Detailed Breakdown and Classification

Green for prime 13 (only 1,13); red for composite 15 (1,3,5,15) with factor arrows.

Visualize divisors around the number: primes have exactly two, composites more. This diagram clarifies why √n testing suffices—one factor ≤ √n implies the pair > √n. Apply to SSAT options by listing quick divisors. Your pattern recognition will shine in timed sections.

SECTION 6

Worked Example: Is 91 Prime?

Determine if 91 is prime or composite

Step 1: Check basics91 odd, not ÷2; doesn't end in 0/5, so not ÷5. Sum 9+1=10 not ÷3.

Step 2: √91 ≈ 9.5, check primes ≤7÷7: 7×13=91? Yes, 91=7×13.

91 composite

Step 3: ConfirmDivisors include 1,7,13,91. Stops early via divisibility.

This example shows elegant shortcut over brute force. SSAT rewards stopping at first divisor. Practice on larger numbers builds confidence. You handle these swiftly now.

SECTION 7

Prime vs Composite: Strengths and Uses

Compare classifications for strategic test-taking.

Type	Properties	SSAT Applications
Prime	Infinite, indivisible >1.	Identify in lists; GCF/LCM.
Composite	Factorable into primes.	Quick elimination; probability.
Neither (1)	Unit, no prime factors.	Distractor in choices.

✦ KEY TAKEAWAY

Primes power cryptography like RSA encryption in apps you use daily; composites enable efficient coding.

SECTION 8

Connections to Advanced Theory

Basic Concept	Advanced Extension
Prime testing up to √n.	Primality tests (AKS algorithm, probabilistic Miller-Rabin).
Unique factorization.	Fundamental Theorem of Arithmetic; RSA security.
Sieve for small n.	Segmented sieve for huge ranges in computing.

SSAT primes preview number theory used in secure online banking. Mastering basics positions you for deeper math success. These foundations unlock puzzles like Goldbach conjecture. You're building a strong quantitative toolkit.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

Which statement best defines a prime number? (A) Divisible only by 1 and itself. (B) Any even number greater than 2. (C) Has exactly three divisors. (D) Sum of two primes. (E) Less than 10.

PROBLEM 2 — BASIC CALCULATION

Is 17 prime or composite? (A) Prime (B) Composite (C) Neither (D) Even (E) Divisible by 3

PROBLEM 3 — INTERMEDIATE

Which is composite? (A) 43 (B) 59 (C) 77 (D) 89 (E) 97

PROBLEM 4 — APPLIED

A code uses primes <20 for security. How many options? (A) 2 (B) 4 (C) 6 (D) 8 (E) 10

PROBLEM 5 — CRITICAL THINKING

If n>1 even, n composite unless? Multi-step: probability n prime if random 100-200? Approx. (A) 0% (B) 5% (C) 10% (D) 25% (E) 50%

SUMMARY

Lesson Summary

Master prime (no divisors 1<d<n) vs composite (has such d); test ≤√n with divisibility rules. Sieve visualizes efficiently; 1 neither.

Apply sieve, divisor checks for SSAT speed. You're equipped to ace quantitative sections confidently.