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Quizzes > High School Quizzes > Mathematics

Prime Factorization Practice Quiz

Sharpen your skills with engaging practice problems

Difficulty: Moderate
Grade: Grade 7
Study OutcomesCheat Sheet
Colorful paper art promoting Prime Factor Frenzy, a math trivia quiz for middle school students.

What is the prime factorization of 12?
2 x 2 x 3
2 x 3 x 3
2 x 6
3 x 4
The prime factors of 12 are 2, 2, and 3 because 12 = 4 x 3 and 4 = 2 x 2. Therefore, the correct factorization is 2 x 2 x 3.
Which of the following is a prime number?
4
6
7
9
The number 7 has only two distinct positive divisors: 1 and itself, making it prime. The other options have additional factors and are composite.
What is the prime factorization of 30?
2 x 3 x 5
2 x 15
3 x 10
5 x 6
30 can be factored by breaking it into 2 x 15, and further factoring 15 as 3 x 5. Thus, the correct prime factorization is 2 x 3 x 5.
Which of the following is the correct prime factorization of 20?
2 x 2 x 5
2 x 5
4 x 5
2 x 10
20 can be expressed as 2 x 10, but 10 is not prime. Breaking it down properly, 20 = 2 x 2 x 5, which is the correct prime factorization.
Simplify: 2 x 3 x 2 x 3. What is its prime factorization?
2 x 2 x 3 x 3
2 x 3
2 x 2 x 3
3 x 3
Multiplying 2 x 3 x 2 x 3 shows that the product is 2^2 x 3^2, which when expanded is 2 x 2 x 3 x 3. This is the correct prime factorization.
Find the prime factorization of 18.
2 x 3 x 3
2 x 9
3 x 6
2 x 2 x 3
18 can be divided into 2 and 9, and since 9 equals 3 x 3, its prime factorization is 2 x 3 x 3.
Which of the following represents the prime factorization of 45?
3 x 3 x 5
5 x 9
3 x 15
2 x 3 x 5
45 breaks down into 9 and 5, and with 9 being 3 x 3, the prime factors are 3, 3, and 5. This makes 3 x 3 x 5 the correct expression.
Determine the prime factorization of 56.
2 x 2 x 2 x 7
2 x 2 x 7
2 x 7 x 7
4 x 14
By repeatedly dividing 56 by 2, you get 2 x 2 x 2, and the remaining factor is 7. Thus, the prime factorization is 2 x 2 x 2 x 7.
Which combination correctly represents the prime factorization of 72?
2 x 2 x 2 x 3 x 3
2 x 2 x 3 x 3
2 x 3 x 3 x 4
2 x 2 x 2 x 2 x 3
72 factors into 2^3 x 3^2, which can be written as 2 x 2 x 2 x 3 x 3. The other options either omit or improperly combine the primes.
What is the prime factorization of 100?
2 x 2 x 5 x 5
2 x 5 x 10
4 x 25
2 x 2 x 5
Breaking 100 as 2 x 50, then 50 as 2 x 25 and 25 as 5 x 5, the prime factorization is 2 x 2 x 5 x 5.
If a number is expressed as 2^3 x 3 x 7, what is the original number?
168
84
336
72
Calculating 2^3 as 8, then multiplying 8 by 3 gives 24, and 24 by 7 results in 168. Hence, the number is 168.
Which of the following numbers cannot be expressed as a product of two or more prime numbers?
13
15
18
21
Composite numbers can always be written as a product of prime factors. Since 13 is prime, it cannot be factored further into two or more primes.
What is the prime factorization of 63?
3 x 3 x 7
3 x 7
7 x 9
3 x 3 x 3
63 can be split into 7 and 9, and since 9 is 3 x 3, the prime factorization is 3 x 3 x 7.
Express 84 as a product of its prime factors.
2 x 2 x 3 x 7
2 x 3 x 7
3 x 3 x 2 x 7
2 x 2 x 7
84 factors as 2 x 42, with 42 further breaking down to 2 x 21 and 21 to 3 x 7. This yields the prime factorization 2 x 2 x 3 x 7.
How many distinct prime factors does 210 have?
4
3
5
6
210 can be factored as 2 x 3 x 5 x 7. Since these are all different prime numbers, there are 4 distinct prime factors.
Find the prime factorization of 252.
2 x 2 x 3 x 3 x 7
2 x 3 x 3 x 7
2 x 2 x 3 x 7 x 7
2 x 2 x 3 x 3 x 3
By successively dividing 252 by 2 and 3, the prime factors are found to be 2, 2, 3, 3, and 7, which corresponds to 2^2 x 3^2 x 7.
If the least common multiple (LCM) of two numbers is 180 and one of the numbers is 12, what prime factorization must the other number have?
3 x 3 x 5
2 x 3 x 5
2 x 2 x 3 x 5
3 x 5
The LCM 180 factors as 2^2 x 3^2 x 5. Since 12 is 2^2 x 3, the missing factors (an extra 3 and a 5) must come from the other number, resulting in 3 x 3 x 5.
Determine the greatest common divisor (GCD) of 84 and 126 using their prime factorizations.
42
14
21
28
84 factors as 2^2 x 3 x 7 and 126 as 2 x 3^2 x 7. The GCD is found by multiplying the lowest powers of the common primes: 2 x 3 x 7, which equals 42.
Suppose a number N is divisible by 8 and has exactly two distinct prime factors, with the exponent of 2 as small as possible. Which of the following is a valid prime factorization for N?
2 x 2 x 2 x 3
2 x 2 x 2 x 3 x 3
2 x 3 x 3
2 x 2 x 3
Since N must be divisible by 8, it needs at least 2^3. With only one other distinct prime factor and the smallest exponent possible for that prime (which is 1), the correct factorization is 2^3 x 3.
Given that the product of two consecutive even numbers is 168, use prime factorization to determine the smaller number.
12
10
14
8
The prime factorization of 168 is 2^3 x 3 x 7. Testing consecutive even numbers shows that 12 and 14 multiply to give 168, so the smaller number is 12.
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Study Outcomes

  1. Understand the definition of prime numbers and their properties.
  2. Identify and explain the process of prime factorization.
  3. Apply prime factorization techniques to decompose composite numbers.
  4. Analyze problems to determine their prime factors efficiently.
  5. Evaluate the accuracy of prime factorization in given mathematical expressions.

Quiz on Prime Factorization Cheat Sheet

  1. Identify Prime Numbers - Prime numbers are natural numbers greater than 1 that have no divisors other than 1 and themselves, making them the "building blocks" of all numbers. Practicing with small numbers like 2, 3, 5, and 7 helps you spot primes quickly. Discover more about primes
  2. Spot Composite Numbers - Composite numbers have more than two positive divisors, so you can break them down into smaller factors. Try dividing numbers like 4, 6, and 8 to see how they split up - it's like solving a mini puzzle! Learn about composites
  3. Master Prime Factorization - Prime factorization is expressing a composite number as a product of primes, for example 28 = 2 × 2 × 7. This skill is essential for simplifying fractions and solving higher‑level problems. Practice factorization
  4. Apply the Fundamental Theorem of Arithmetic - This theorem tells us every integer greater than 1 has a unique prime factorization, no matter the order of primes. Knowing this rule ensures you never miss or duplicate factors in your breakdowns. Read about the theorem
  5. Use the Factor Tree Method - Draw a tree diagram to break a number into two factors, then keep splitting until all branches are primes. For 60, you'd go 60 → 2 × 30 → 2 × 15 → 3 × 5, giving 2 × 2 × 3 × 5. See factor trees in action
  6. Try the Birthday Cake (Ladder) Method - This stepwise division method stacks prime divisors like layers on a cake: divide by 2, then by the next prime, and so on. It's a neat visual trick that many students find fun and efficient! Explore the ladder method
  7. Link to GCD and LCM - Prime factorization makes finding the greatest common divisor (GCD) and least common multiple (LCM) a breeze - you just compare prime exponents. This unlocks shortcuts for adding, subtracting, and comparing fractions. Connect primes to GCD & LCM
  8. Discover Cryptography Connections - Modern encryption methods like RSA depend on the difficulty of factoring huge composite numbers into primes. Learning prime factorization gives you a peek behind the curtain of secure online communication. Learn why primes matter in crypto
  9. Practice with Worksheets and Booklets - Regular drills build speed and confidence - mix easy, medium, and challenge numbers to keep your brain sharp. Printable practice booklets turn study time into a game with levels to conquer! Grab your practice booklet
  10. Remember the Special Case of 1 - Number 1 is neither prime nor composite because it has only one positive divisor (itself). Always exclude 1 when you start prime factorization to avoid common mistakes. Why 1 is unique
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