Prime and Composite Numbers — Determine whether a number is prime or composite. - SHSAT Math

SECTION 1

Historical Context & Motivation

People have been fascinated by numbers for thousands of years. Ancient mathematicians noticed something interesting: some numbers can only be divided evenly by 1 and themselves, while others have many divisors. These special "unbreakable" numbers became known as prime numbers. Understanding primes has shaped the way we do math, build codes, and even protect information online.

~300 BCE • Euclid's Elements

The Greek mathematician Euclid proved that there are infinitely many prime numbers. He also showed that every whole number can be built from primes.

~240 BCE • The Sieve of Eratosthenes

Eratosthenes invented a simple method (called a "sieve") for finding all prime numbers up to any limit. This technique is still taught today!

1801 • Gauss and Number Theory

Carl Friedrich Gauss published a famous book that proved the Fundamental Theorem of Arithmetic: every integer greater than 1 can be written as a unique product of primes.

1970s–Today • Primes in Computer Security

Modern encryption (the technology that keeps your passwords safe) relies on the fact that multiplying two huge primes is easy, but figuring out which primes were used is extremely hard.

So how do we actually decide whether a number is prime or composite? That's exactly what this lesson will teach you — step by step.

SECTION 2

Core Principles & Definitions

Before we start testing numbers, let's lock down the key vocabulary. Every SHSAT question about primes and composites relies on these ideas.

Factor (Divisor)

A factor is a whole number that divides into another number with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.

Prime Number

A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13.

Composite Number

A composite number is a whole number greater than 1 that has more than two factors. Examples: 4, 6, 8, 9, 10, 12.

The Number 1

The number 1 is neither prime nor composite. It only has one factor (itself), so it doesn't meet the definition of either category.

The Only Even Prime

The number 2 is the only even prime number. Every other even number can be divided by 2, so they all have at least three factors.

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KEY TAKEAWAY

Think of a prime number like an atom — it's a basic building block that can't be split into smaller pieces. A composite number is like a molecule: you can break it apart into smaller prime "atoms." On the SHSAT, the fastest way to check is to ask: "Does any whole number besides 1 and the number itself divide into it evenly?" If yes, it's composite. If no, it's prime.

SECTION 3

Visual Explanation — The Sieve of Eratosthenes

The best way to see primes and composites is with a number grid. The diagram below shows the numbers 1 through 50. Primes are highlighted in bright colors, composites are dimmed, and the number 1 is marked separately. Notice how primes become less common as numbers get bigger.

Numbers 1 through 50 in a 10-column grid. Brightly colored cells are prime numbers, dimmed cells are composite numbers, and the gray cell (1) is neither. There are 15 primes in this range.

Look at the grid above. The bright cells are scattered among the dim ones, and there's no neat pattern to where primes appear. That's actually one of the most interesting things about them. However, you can spot some quick rules: after the number 2, no even number is prime. After 5, no number ending in 0 or 5 is prime.

SECTION 4

The Divisibility Test Method

You don't need to check every possible divisor to decide if a number is prime. There's a shortcut. You only need to test divisors up to the square root of the number. Here's why: if a number has a factor bigger than its square root, then it must also have a matching factor that's smaller than the square root. So if no small factor works, no big factor will either!

SQUARE ROOT TEST RULE

To test whether n is prime, check divisibility by every prime ≤ √n

n = the number you're testing. √n = the square root of n. You only need to try prime divisors (2, 3, 5, 7, 11, …) up to that square root.

For most numbers on the SHSAT (typically under 100), you only need to check divisibility by 2, 3, 5, and 7. That's because √100 = 10, and the primes up to 10 are 2, 3, 5, and 7. Here are the quick divisibility rules to memorize:

DIVISIBLE BY 2

A number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8.

Example: 84 ends in 4, so 84 ÷ 2 = 42. It's composite.

DIVISIBLE BY 3

A number is divisible by 3 if the sum of its digits is divisible by 3.

Example: 51 → 5 + 1 = 6, and 6 ÷ 3 = 2. So 51 is divisible by 3 and is composite.

DIVISIBLE BY 5

A number is divisible by 5 if its last digit is 0 or 5.

Example: 85 ends in 5, so 85 ÷ 5 = 17. It's composite.

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SHSAT TIP

On test day, start with the easiest check first. Is the number even? If yes (and it's not 2), it's composite — done! Next, add the digits to test for 3. Then check the last digit for 5. Finally, try dividing by 7. This order saves you the most time.

SECTION 5

Factor Trees & Prime Factorization

A factor tree is a diagram that breaks a composite number down into its prime factors. You keep splitting until every "leaf" of the tree is a prime number. This is called prime factorization. It's a powerful way to confirm that a number is composite and to see exactly which primes make it up.

Left: A factor tree for 60, showing how it breaks down into 2 × 2 × 3 × 5. Orange nodes can still be split; colored leaf nodes are primes. Right: Testing 37, which cannot be split — it's prime.

On the left side of the diagram, 60 gets split over and over until only primes remain. On the right, 37 can't be split at all because none of the primes up to its square root divide into it evenly. This comparison shows the key difference between composite and prime numbers in action.

Number	√n (approx.)	Primes to Test	Result
29	5.39	2, 3, 5	Prime
51	7.14	2, 3, 5, 7	Composite (3 × 17)
67	8.19	2, 3, 5, 7	Prime
91	9.54	2, 3, 5, 7	Composite (7 × 13)

Applying the square root test to several numbers

SECTION 6

Worked Example — Is 83 Prime or Composite?

Let's walk through a full example, step by step. This is exactly the process you should use on the SHSAT when a question asks you to identify whether a number is prime or composite.

Is 83 prime or composite?

1
Step 1 — Find the Square Root

Calculate √83. Since 9 × 9 = 81 and 10 × 10 = 100, we know √83 is between 9 and 10. So we only need to test prime divisors up to 9.

√83 ≈ 9.1 → test primes: 2, 3, 5, 7
2
Step 2 — Test Divisibility by 2

Is 83 even? The last digit is 3, which is odd. So 83 is not divisible by 2.

83 ÷ 2 = 41.5 ✗
3
Step 3 — Test Divisibility by 3

Add the digits: 8 + 3 = 11. Is 11 divisible by 3? No (11 ÷ 3 = 3.67). So 83 is not divisible by 3.

83 ÷ 3 = 27.67… ✗
4
Step 4 — Test Divisibility by 5

Does 83 end in 0 or 5? No, it ends in 3. So 83 is not divisible by 5.

83 ÷ 5 = 16.6 ✗
5
Step 5 — Test Divisibility by 7

Divide: 83 ÷ 7 = 11.86 (not a whole number). So 83 is not divisible by 7.

83 ÷ 7 = 11.86… ✗
6
Step 6 — Conclusion

We tested every prime up to √83, and none of them divided evenly. Therefore, 83 is a prime number.

83 is PRIME

SECTION 7

Common Traps & Quick Tips

The SHSAT loves to test tricky numbers that look prime but aren't (or look composite but are). Let's compare some common mistakes students make with the correct thinking.

Common Trap	Why It's Wrong	Correct Thinking
"1 is prime because its only factor is itself."	A prime must have exactly TWO factors. The number 1 has only ONE factor.	1 is neither prime nor composite.
"All odd numbers are prime."	Many odd numbers are composite: 9, 15, 21, 25, 27, 33, 35, 39, etc.	Always test — odd doesn't mean prime.
"2 isn't prime because it's even."	2 has exactly two factors (1 and 2), which fits the prime definition perfectly.	2 is prime — the only even prime.
"91 is prime because it looks prime."	91 = 7 × 13. Students forget to test 7 as a divisor.	Always test ALL primes up to √n.
"A big number can't be prime."	Large primes exist everywhere: 97, 101, 103, 107, etc.	Size doesn't determine primality — factors do.

Common student mistakes on prime/composite questions

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WATCH OUT FOR THESE SHSAT FAVORITES

The numbers 51, 57, 87, and 91 are commonly mistaken for primes on tests. Quick checks: 51 = 3 × 17, 57 = 3 × 19, 87 = 3 × 29, and 91 = 7 × 13. Memorizing these can save you valuable time.

SECTION 8

Connecting to Bigger Ideas

Understanding primes and composites is the foundation for many harder topics you'll see on the SHSAT and in future math courses. Here's how this concept connects to what comes next.

This Lesson	Where It Leads
Identifying prime numbers	Finding the Greatest Common Factor (GCF) of two numbers by comparing their prime factorizations
Prime factorization	Finding the Least Common Multiple (LCM) using prime factors
Divisibility rules	Simplifying fractions — divide top and bottom by common prime factors
Counting factors of a number	SHSAT questions that ask "how many factors does n have?" — use prime factorization to count systematically

How primes connect to other SHSAT topics

Once you're comfortable identifying primes, try this: pick any composite number and write it as a product of primes. Then use that skill to find the GCF or LCM of two numbers. You'll see how primes are the "master key" for many SHSAT number-property questions.

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PRIME NUMBERS TO MEMORIZE FOR THE SHSAT

Primes under 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. That's 25 primes total. Knowing this list cold will help you answer questions in seconds.

SECTION 9

Practice Problems

Try these five problems. They start easy and get harder, just like the SHSAT. Work through each one using the square root test before checking the answer.

Question 1: CONCEPTUAL

True or false: The number 1 is a prime number. Explain your reasoning.

Show answer

False. A prime number must have exactly two factors: 1 and itself. The number 1 has only one factor (which is 1). Since it doesn't meet the definition of having exactly two distinct factors, 1 is classified as neither prime nor composite. This is a very common SHSAT trap!

Question 2: BASIC CALCULATION

Is 57 prime or composite? Show your work using divisibility rules.

Show answer

57 is composite. First, check √57 ≈ 7.5, so test primes 2, 3, 5, and 7. The number 57 is odd (not divisible by 2). Add the digits: 5 + 7 = 12, and 12 ÷ 3 = 4, so 57 IS divisible by 3. We can confirm: 57 ÷ 3 = 19. Since 57 = 3 × 19, it has more than two factors, making it composite.

Question 3: INTERMEDIATE

Which of the following numbers are prime: 41, 49, 53, 63, 71? List all the primes and explain how you know.

Show answer

The primes are 41, 53, and 71. For each, test primes up to the square root. 49 = 7 × 7 (composite). 63: digits sum to 9, which is divisible by 3, so 63 = 3 × 21 (composite). For 41: √41 ≈ 6.4, test 2, 3, 5 — none divide evenly. For 53: √53 ≈ 7.3, test 2, 3, 5, 7 — none work. For 71: √71 ≈ 8.4, test 2, 3, 5, 7 — none work. So 41, 53, and 71 pass all tests and are prime.

Question 4: APPLIED

A teacher wants to divide 97 students into equal groups with more than 1 student per group and more than 1 group. Is this possible? Explain using what you know about prime and composite numbers.

Show answer

No, it is not possible. The number 97 is prime. To test: √97 ≈ 9.8, so check 2, 3, 5, and 7. 97 is odd (not div. by 2), 9 + 7 = 16 (not div. by 3), doesn't end in 0 or 5 (not div. by 5), 97 ÷ 7 = 13.86 (not div. by 7). Since 97 has only two factors (1 and 97), the only ways to make equal groups are 1 group of 97 or 97 groups of 1 — neither satisfies both conditions.

Question 5: CRITICAL THINKING

Twin primes are pairs of prime numbers that differ by 2, like (3, 5) or (11, 13). Find all twin prime pairs where both numbers are between 30 and 80. Explain your process.

Show answer

The twin prime pairs between 30 and 80 are (41, 43), (59, 61), and (71, 73). First, list the primes between 30 and 80: 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79. Now check which consecutive primes differ by exactly 2. 41 and 43 differ by 2 ✓. 59 and 61 differ by 2 ✓. 71 and 73 differ by 2 ✓. No other consecutive primes in this range are exactly 2 apart. This problem previews an advanced idea — mathematicians still don't know if twin primes go on forever!

SUMMARY

Lesson Summary

Summary

A prime number is a whole number greater than 1 with exactly two factors (1 and itself), while a composite number has more than two factors. The number 1 is neither prime nor composite, and 2 is the only even prime. To test whether a number is prime, use the square root test: check divisibility by every prime up to the square root of the number.

For SHSAT numbers under 100, you only need to test 2, 3, 5, and 7 using divisibility rules. Watch out for tricky composites like 51, 57, 87, and 91. Mastering primes unlocks bigger topics like GCF, LCM, and fraction simplification — all of which appear frequently on the SHSAT.