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Prime & Composite
Numbers

Learn to identify whether a number is prime or composite by analyzing its factors: a foundational skill for the SSAT and beyond.

SSAT Middle Level

What Is a Factor?

A factor of a number is a whole number that divides it evenly, with no remainder. Every whole number greater than \(1\) has at least two factors: \(1\) and the number itself.

What Is a Prime Number?

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

\(7 \text{ is prime}\)
\(\text{Factor 1: } 1\)
\(\text{Factor 2: } 7\)
\(\text{Only 2 factors } \rightarrow \text{ Prime}\)

What Is a Composite Number?

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

\(12 \text{ is composite}\)
\(1\)
\(2\)
\(3\)
\(4\)
\(6\)
\(12\)
\(\text{6 factors } \rightarrow \text{ Composite}\)

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

\(0\)
Zero is neither prime nor composite. Every number is a factor of \(0\), so it does not meet the definition of either category.
\(1\)
One is neither prime nor composite. It has only one factor (itself), but primes require exactly two distinct factors.

The Core Rule

\(\text{Number } n > 1\)
\(\text{Does any number from } 2 \text{ to } n-1\)
\(\text{divide } n \text{ evenly?}\)
 No Yes Prime Composite
Efficiency shortcut
You only need to test divisors up to the \(\sqrt{n}\). If none divide \(n\), it is prime.

Visualizing Factors with Arrays

One way to see whether a number is prime or composite is to try arranging that many dots into rectangular arrays. Primes can only form a single row.

\(7 \text{ (Prime)}\)
\(\text{Only } 1 \times 7\)
\(12 \text{ (Composite)}\)
\(2 \times 6\)
\(3 \times 4\)
Only one rectangle Multiple rectangles

Prime vs Composite

Prime Composite
Number of factors Exactly 2 3 or more
Factors are \(1\) and itself \(1\), itself, and others
Rectangular arrays Only \(1 \times n\) Multiple arrangements
Smallest example \(2\) \(4\)
Even examples Only \(2\) \(4, 6, 8, 10, \ldots\)

Why \(2\) Is Special

Smallest Prime
\(2\) is the smallest prime and the only even prime. Its factors are exactly \(1\) and \(2\).
All Other Evens
Every even number > \(2\) is divisible by \(2\), so it automatically has a third factor, making it composite.

How to Test a Number

1
Check the basics. Is the number \(\leq 1\)? Then it is neither prime nor composite.
2
Is it \(2\) or \(3\)? These are prime. Done.
3
Is it even? If the number is even and greater than \(2\), it is composite.
4
Test odd divisors. Try dividing by \(3, 5, 7, \ldots\) up to \(\sqrt{n}\).
5
Decide. If no divisor works, the number is prime. If any divisor works, it is composite.

Divisibility Quick Checks

These shortcuts let you quickly spot composite numbers without long division.

Divisible by \(2\)
Last digit is even (\(0, 2, 4, 6, 8\)).
Divisible by \(3\)
Sum of digits is divisible by \(3\).
Divisible by \(5\)
Last digit is \(0\) or \(5\).

Example: Is \(29\) Prime?

1
\(29 > 1\), so it could be prime or composite.
2
\(29\) is odd, so it is not divisible by \(2\).
3
Compute \(\sqrt{29} \approx 5.4\). Test primes up to \(5\): that means \(2, 3, 5\).
4
\(29 \div 3 = 9\text{ R }2\): not divisible. \(29 \div 5 = 5\text{ R }4\): not divisible.
5
No divisor found. \(29\) is prime.

Example: Is \(51\) Prime?

1
\(51 > 1\) and it is odd, so not divisible by \(2\).
2
Digit sum: \(5 + 1 = 6\). Since \(6\) is divisible by \(3\), so is \(51\).
3
\(51 \div 3 = 17\). We found a factor other than \(1\) and \(51\).

Example: Is \(97\) Prime?

1
\(97\) is odd, so not divisible by \(2\).
2
\(\sqrt{97} \approx 9.8\). We test primes: \(2, 3, 5, 7\).
3
Digit sum: \(9 + 7 = 16\). Not divisible by \(3\).
4
Last digit is \(7\), not \(0\) or \(5\): not divisible by \(5\).
5
\(97 \div 7 = 13\text{ R }6\): not divisible by \(7\).

Common Mistakes

✕ Calling \(1\) prime
\(1\) has only one factor, not two. By definition, it is neither prime nor composite.
✕ Forgetting \(2\) is prime
\(2\) is even, but it has exactly two factors (\(1\) and \(2\)). It is the only even prime.
✕ Assuming odd = prime
Many odd numbers are composite: \(9, 15, 21, 25, 27, 33, 35, 49, 51, \ldots\)

Summary Table

Number Factors Classification
\(1\) \(1\) Neither
\(2\) \(1, 2\) Prime
\(9\) \(1, 3, 9\) Composite
\(13\) \(1, 13\) Prime
\(15\) \(1, 3, 5, 15\) Composite
\(23\) \(1, 23\) Prime
\(49\) \(1, 7, 49\) Composite

Question 1 of 5

Which of the following numbers is prime?

Question 2 of 5

What is the smallest prime number?

Question 3 of 5

How many even prime numbers exist?

Question 4 of 5

Which number is composite?

Question 5 of 5

To determine whether \(83\) is prime, up to which number do you need to check for divisors?