Prime and Composite Numbers

🔢 Prime and Composite Numbers

Part 1 of 5 — Factors: The Building Blocks

Topics in This Part

🔑 Key Concept: A factor is a whole number that divides another number evenly — with no remainder. Before we can sort numbers into "prime" and "composite," we first need to be experts at finding factors. That's Part 1.

What Is a Factor?

A factor of a number is a whole number that divides into it evenly — meaning the answer is a whole number and nothing is left over.

Think about $12$ cookies. You can share them into equal groups in several ways:

Every number that fits perfectly is a factor. So the factors of $12$ are:

$1,\; 2,\; 3,\; 4,\; 6,\; 12$

💡 Quick test: A number is a factor if you can divide and get no remainder. $12 \div 5 = 2$ remainder $2$, so $5$ is not a factor of $12$.

Concept Check 🎯

Finding ALL the Factors

To list every factor of a number, work in pairs. Start at $1$ and climb upward, checking each number to see if it divides evenly. Each time it does, you find two factors at once — a factor pair.

Worked Example: Factors of $18$

Once the pairs start to "meet in the middle," you can stop. The factors of $18$ are:

$1,\; 2,\; 3,\; 6,\; 9,\; 18$

💡 Stay organized: Listing factors smallest-to-largest means you will never miss one. Every factor pair multiplies back to the original number.

Count the Factors 🧮

For each number, find how many factors it has in total. (List them in your head smallest to largest, then count.)

1) How many factors does $10$ have? 2) How many factors does $7$ have? 3) How many factors does $16$ have?

Complete the Factor Pair 🧮

Each pair of factors multiplies to make the target number. Fill in the missing partner.

1) A factor pair of $24$ is $3$ and $?$ 2) A factor pair of $36$ is $4$ and $?$ 3) A factor pair of $30$ is $6$ and $?$

🔢 Prime and Composite Numbers

Part 2 of 5 — Prime vs. Composite

🔑 The Big Idea: Now that you can find factors, you can sort whole numbers into two families. The secret is simple: how many factors does the number have?

The Two Families

We sort whole numbers (greater than $1$) by counting their factors:

Prime Numbers

A prime number has exactly two factors: $1$ and the number itself.

• $7$ is prime → factors are just $1$ and $7$.
• $13$ is prime → factors are just $1$ and $13$.

Composite Numbers

A composite number has more than two factors.

• $12$ is composite → factors $1, 2, 3, 4, 6, 12$ (that's $6$ factors).
• $9$ is composite → factors $1, 3, 9$ (that's $3$ factors).

🔑 The rule in one line: Count the factors. Exactly two means prime. Three or more means composite.

Concept Check 🎯

The Two Tricky Numbers: $1$ and $2$

These two numbers confuse a lot of students. Learn them now.

The Number $1$ — Neither Prime nor Composite

The number $1$ has only one factor — just itself, $1$.

• It does not have exactly two factors, so it is not prime.
• It does not have more than two factors, so it is not composite.

$\boxed{1 \text{ is neither prime nor composite.}}$

The Number $2$ — The Only Even Prime

The number $2$ has exactly two factors: $1$ and $2$. That makes it prime.

It is the only even prime number. Every other even number (like $4, 6, 8, \dots$) can be divided by $2$, giving it an extra factor — so all other even numbers are composite.

⚠️ Remember these two facts: $1$ is neither prime nor composite, and $2$ is the only even prime.

Sort Each Number 🔽

Decide whether each number is prime, composite, or neither.

Spot the Prime 🎯

🔢 Prime and Composite Numbers

Part 3 of 5 — Divisibility Rules: Fast Tests

🔑 Why this helps: Instead of slowly dividing, divisibility rules let you tell at a glance whether a number has a small factor. If it has any factor besides $1$ and itself, it must be composite.

The Divisibility Rules

These quick tests tell you whether a number divides evenly by $2$, $3$, $5$, or $10$ — without doing the division.

Worked Example: Is $72$ composite?

• Ends in $2$, so it's divisible by $2$. ✓

Since $2$ is a factor (besides $1$ and $72$), the number $72$ has more than two factors — it is composite.

💡 The key link: If any small number divides evenly, you've found an extra factor — and that instantly makes the number composite.

Apply the Rules 🔽

Use the divisibility rules to answer each one.

How to Test if a Number Is Prime

To check whether a number is prime, try to divide it by the small primes in order: $2, 3, 5, 7, \dots$ If none of them divide evenly, the number is prime.

Worked Example: Is $29$ prime?

• Divisible by $2$? No — $29$ is odd.
• Divisible by $3$? No — $2+9 = 11$, not a multiple of $3$.
• Divisible by $5$? No — doesn't end in $0$ or $5$.

No small number divides it, so $\boxed{29 \text{ is prime.}}$

Worked Example: Is $51$ prime?

• Divisible by $3$? $5 + 1 = 6$, which is a multiple of $3$. ✓

So $51 = 3 \times 17$. It has an extra factor, which means $\boxed{51 \text{ is composite.}}$

⚠️ Watch out for "fake primes." Numbers like $51$, $57$, $91$, and $87$ look prime but secretly have a factor of $3$ or $7$. Always run the divisibility tests!

Concept Check 🎯

Prime or Composite? 🧮

For each number, type P if it is prime or C if it is composite.

1) $19$ 2) $39$ 3) $2$

🔢 Prime and Composite Numbers

Part 4 of 5 — Prime Factorization & Factor Trees

🔑 Big Payoff: Every composite number can be broken down into a unique set of prime building blocks. Writing a number as a product of primes is called its prime factorization.

Building a Factor Tree

A factor tree breaks a number down step by step until only primes are left.

Worked Example: Prime factorization of $36$

Split $36$ into any factor pair, then keep splitting until every "leaf" is prime:

$36 = 4 \times 9$ $4 = 2 \times 2 \qquad 9 = 3 \times 3$

The leaves at the bottom are $2, 2, 3, 3$ — all primes. So:

$36 = 2 \times 2 \times 3 \times 3$

Using exponents, we write this neatly as:

$36 = 2^2 \times 3^2$

💡 Same answer every time: It doesn't matter which factor pair you start with. Starting $36 = 6 \times 6$ gives the exact same primes: $2 \times 2 \times 3 \times 3$. This is called unique prime factorization.

Worked Example: Prime factorization of $60$

$60 = 6 \times 10$ $6 = 2 \times 3 \qquad 10 = 2 \times 5$

The prime leaves are $2, 3, 2, 5$. Writing them smallest to largest:

$60 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5$

✅ Check: $2 \times 2 \times 3 \times 5 = 4 \times 15 = 60$ ✓

🔑 Tip: A factor tree is finished only when every number at the bottom is prime. If you can still split a number, keep going!

Finish the Factor Tree 🔽

You are finding the prime factorization of $24$. Choose what happens at each step.

Count the Prime Factors 🧮

Build a factor tree, then count how many prime numbers are at the bottom (count repeats).

1) In $2^2 \times 3^2$ (which is $36$), how many prime factors are there in total? 2) The prime factorization of $8$ is $2 \times 2 \times 2$. How many prime factors? 3) The prime factorization of $30$ is $2 \times 3 \times 5$. How many prime factors?

Fill In the Factorization 🧮

Each number's prime factorization is started for you. Find the missing prime.

1) $18 = 2 \times 3 \times ?$ 2) $20 = 2 \times 2 \times ?$ 3) $42 = 2 \times 3 \times ?$

🔢 Prime and Composite Numbers

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) find all the factors of a number, (2) sort numbers into prime and composite, (3) use divisibility rules to test for primes, and (4) build a prime factorization. Let's put it all together.

Quick Reference

Primes to know by heart (under $50$):

$2,\; 3,\; 5,\; 7,\; 11,\; 13,\; 17,\; 19,\; 23,\; 29,\; 31,\; 37,\; 41,\; 43,\; 47$

⚠️ Don't forget: Being odd does not make a number prime ($9$, $15$, $21$, $25$ are all odd and composite), and the digit-sum test for $3$ catches sneaky composites like $51$ and $57$.

Mixed Practice 🎯

Final Drill 🧮

1) How many factors does the prime number $13$ have? 2) The prime factorization of $100$ is $2 \times 2 \times 5 \times 5$. Written with exponents it is $2^2 \times 5^?$ — what exponent goes on the $5$? 3) What is the only even prime number?

Exit Quiz ✅

Answer all three to finish the lesson.