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Prime & Composite
Numbers

Master how to identify and classify numbers as prime or composite: the building blocks of all number theory.

SSAT Upper Level: Quantitative

What Is a Factor?

A factor of a number \( n \) is any whole number that divides \( n \) evenly, meaning the remainder is \( 0 \). Understanding factors is the key to classifying numbers.

Defining Prime Numbers

A prime number is a whole number greater than \( 1 \) that has exactly two distinct factors: \( 1 \) and itself. No other number divides it evenly.

First 8 Prime Numbers 2 3 5 7 11 13 17 19

Defining Composite Numbers

A composite number is a whole number greater than \( 1 \) that has more than two distinct factors. It can be divided evenly by at least one number other than \( 1 \) and itself.

Factor Tree of 12 12 2 6 2 3
\( = 2 \times 2 \times 3 \)

The Special Cases: \( 0 \) and \( 1 \)

\( 0 \)
Zero is neither prime nor composite. It is divisible by every nonzero integer, so it has infinitely many factors, which breaks both definitions.
\( 1 \)
Neither prime nor composite. It has only one factor (itself). Prime requires exactly two, and composite requires more than two.

The Fundamental Theorem of Arithmetic

60 = 2 × 2 × 3 × 5
Compact form: \( 2^2 \times 3 \times 5 \)

This means prime numbers are the building blocks of all integers: every composite number is built by multiplying primes together.

Prime vs. Composite

Prime Composite
Factor count Exactly \( 2 \) More than \( 2 \)
Smallest example \( 2 \) \( 4 \)
Even numbers? Only \( 2 \) \( 4, 6, 8, 10, \dots \)
Divisible by \( 1 \) and itself only At least one other number
Role in math Building blocks Products of primes

Divisibility Rules Shortcut

Before testing every possible factor, use divisibility rules to quickly check if a number is composite.

2
Divisible by \( 2 \): The number is even (last digit is \( 0, 2, 4, 6, \) or \( 8 \)).
3
Divisible by \( 3 \): The sum of its digits is divisible by \( 3 \).
5
Divisible by \( 5 \): The last digit is \( 0 \) or \( 5 \).
7
Divisible by \( 7 \): Double the last digit, subtract from the rest. If the result is divisible by \( 7 \), so is the original.

The Square Root Test

Why? If \( n = a \times b \), then at least one of \( a \) or \( b \) must be \(\leq \sqrt{n}\). So if no small factor exists, no factor pair exists at all.

How to Classify a Number

1
Check the basics: Is the number \( \leq 1 \)? If yes, it is neither prime nor composite. Stop.
2
Apply divisibility rules: Test by \( 2, 3, 5 \). If any divides the number and it is not equal to that prime, it is composite.
3
Find \( \sqrt{n} \): Only test primes up to this value. Primes to check: \( 2, 3, 5, 7, 11, 13, \dots \)
4
Divide systematically: If any prime divides \( n \) evenly, it is composite. If none do, it is prime.

Example: Is \( 51 \) Prime?

1
Is \( 51 > 1 \)? Yes. Continue.
2
Divisible by \( 2 \)? \( 51 \) is odd. No.
3
Divisible by \( 3 \)? Digit sum: \( 5 + 1 = 6 \). Since \( 6 \div 3 = 2 \), yes!
4
Confirm: \( 51 \div 3 = 17 \). Found a factor other than \( 1 \) and \( 51 \).

Example: Is \( 67 \) Prime?

1
Compute \( \sqrt{67} \approx 8.19 \). Test primes \( \leq 8 \): that's \( 2, 3, 5, 7 \).
2
By \( 2 \): \( 67 \) is odd. No.
3
By \( 3 \): \( 6 + 7 = 13 \). Not divisible by \( 3 \). No.
4
By \( 5 \): Doesn't end in \( 0 \) or \( 5 \). No.
5
By \( 7 \): \( 67 \div 7 = 9.57\dots \) Not whole. No.

Example: Is \( 91 \) Prime?

This is a classic trap. The number \( 91 \) looks prime, but let's apply our method carefully.

1
\( \sqrt{91} \approx 9.54 \). Test primes \( \leq 9 \): \( 2, 3, 5, 7 \).
2
By \( 2 \): Odd. By \( 3 \): \( 9+1=10 \), no. By \( 5 \): Ends in \( 1 \), no.
3
By \( 7 \): \( 91 \div 7 = 13 \). It divides evenly!

The Sieve of Eratosthenes

An ancient algorithm for finding all primes up to a given limit: cross out multiples of each prime, starting from \( 2 \).

Sieve for Numbers 2–26 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Common Mistakes

✕ Calling \( 1 \) prime
\( 1 \) has only one factor, not two. By definition, prime requires exactly two distinct factors.
✕ Assuming odd = prime
Many odd numbers are composite: \( 9, 15, 21, 25, 27, 33, 35, 49, 51 \dots \) Always test factors.
✕ Stopping at \( 5 \)
For larger numbers, you must also test \( 7, 11, 13 \dots \) up to \( \sqrt{n} \). Missing \( 7 \) leads to errors on \( 49, 77, 91 \).

Quick Reference Summary

Number Classification Factors
\( 1 \) Neither \( 1 \)
\( 2 \) Prime \( 1, 2 \)
\( 9 \) Composite \( 1, 3, 9 \)
\( 29 \) Prime \( 1, 29 \)
\( 51 \) Composite \( 1, 3, 17, 51 \)
\( 97 \) Prime \( 1, 97 \)

Question 1 of 5

Which of the following numbers is prime?

Question 2 of 5

Which statement about the number \( 1 \) is correct?

Question 3 of 5

To determine whether \( 113 \) is prime, you need to check divisibility by all primes up to:

Question 4 of 5

How many prime numbers are there between \( 20 \) and \( 40 \) inclusive?

Question 5 of 5

Which of the following is the prime factorization of \( 84 \)?