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Quizzes > Mathematics

Number Theory Quiz: Prime Factors and Greatest Common Factor

Quick, free prime factorization quiz with GCF questions. Instant results.

Editorial: Review CompletedCreated By: Allan EdmondUpdated Aug 25, 2025
Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper cut numbers and symbols on blue background free number theory quiz prime factorization gcd algebraic factorization

This quiz helps you practice prime factorization, factor pairs, and finding the greatest common factor to spot gaps before class or a test. Want more focused work? Try the greatest common factor quiz, explore a broader number theory quiz, or review basics with a factors and multiples quiz.

What is the prime factorization of 36?
2^4 * 3^0
2^2 * 3^2
2 * 3^3
2^3 * 3
The prime factorization of 36 breaks down to 2 · 2 · 3 · 3, which is 2^2 · 3^2. Every integer greater than 1 can be uniquely expressed as a product of primes according to the Fundamental Theorem of Arithmetic. Recognizing repeated primes gives the exponent form.
Which of the following is a prime number?
15
29
21
49
A prime number has exactly two distinct positive divisors: 1 and itself. Among the options, 29 is not divisible by any integer other than 1 and 29. Numbers like 15, 21, and 49 have additional divisors.
Find the greatest common divisor (GCD) of 18 and 24.
6
12
3
9
The divisors of 18 are 1, 2, 3, 6, 9, 18; of 24 are 1, 2, 3, 4, 6, 8, 12, 24. The largest common divisor is 6. The Euclidean algorithm also yields 6 quickly.
How many distinct prime factors does 30 have?
4
5
2
3
The number 30 factors to 2 · 3 · 5, which are three distinct primes. Counting each unique prime gives the answer. Exponents are not considered distinct if the prime appears only once.
What is the smallest prime factor of 91?
13
7
3
11
Dividing 91 by 7 yields 13, which shows 91 = 7 · 13. Since 7 is smaller than 13 and is prime, it's the smallest prime factor. You test divisibility starting from the smallest prime.
True or False: 1 is a prime number.
True
False
By definition, a prime number has exactly two distinct positive divisors. The number 1 has only one divisor (itself), so it is not prime. This is a fundamental convention in number theory.
What is the least common multiple (LCM) of 4 and 6?
24
8
12
6
The LCM of two numbers is the smallest positive integer divisible by both. Prime factorizations: 4 = 2^2, 6 = 2 · 3, so take 2^2 · 3 = 12. This yields the least multiple containing both prime requirements.
Which number is composite?
29
37
33
31
A composite number has more than two positive divisors. The number 33 = 3 · 11, which makes it composite. The others listed are primes with only two divisors.
Find the greatest common divisor (GCD) of 56 and 98.
28
7
14
56
Using prime factorizations: 56 = 2^3 · 7, 98 = 2 · 7^2. The GCD takes the minimum exponent for each prime: 2^1 · 7^1 = 14. The Euclidean algorithm also confirms this.
What is the prime factorization of 84?
2^2 * 3 * 7
2^3 * 3 * 5
2 * 3^2 * 7
2 * 7^2
Break 84 into primes: 84 = 2 · 42 = 2 · 2 · 21 = 2^2 · 3 · 7. This matches the selected answer. Each step divides out the smallest possible prime.
Factor x^2 - 9.
(x - 2)(x + 7)
(x - 3)(x + 3)
(x - 1)(x + 9)
(x - 9)(x + 1)
x^2 - 9 is a difference of squares: a^2 - b^2 = (a - b)(a + b) with a = x and b = 3. This factors directly into (x - 3)(x + 3).
Factor 15x^2y - 35xy.
xy(15x - 35)
5xy(3x - 7)
5x(3x - 7)y^2
15xy(x - 7)
Both terms share the common factor 5xy. Factoring that out leaves 3x - 7. So the expression is 5xy(3x - 7). Factoring out the greatest common monomial simplifies the expression.
What is the sum of the exponents in the prime factorization of 512?
9
8
7
10
512 = 2^9, as dividing repeatedly by 2 yields nine factors of 2. Summing the single exponent gives 9. Recognizing powers of two helps quickly identify the exponent.
Which relationship holds for any positive integers a and b, given their GCD and LCM?
ab = gcd(a,b) * lcm(a,b)
ab = gcd(a,b) + lcm(a,b)
ab = gcd(a,b) - lcm(a,b)
ab = lcm(a,b) - gcd(a,b)
A key identity in number theory is that the product of two integers equals the product of their GCD and LCM. That is, a · b = gcd(a,b) · lcm(a,b). This follows from prime exponent comparisons.
Which of the following pairs are co-prime?
(18, 27)
(21, 28)
(14, 15)
(8, 12)
Two numbers are co-prime if their GCD is 1. gcd(14,15) = 1, while the other pairs share common divisors larger than 1. Checking prime factors confirms this quickly.
True or False: By the divisibility rule for 11, the number 2728 is divisible by 11.
False
True
The rule for 11 checks the alternating sum of digits: (2 + 2) ? (7 + 8) = 4 ? 15 = -11, which is divisible by 11. Therefore 2728 is divisible by 11. This rule works for any integer in base 10.
What is Euler's totient ?(15)?
7
12
8
4
?(n) counts positive integers ? n that are co-prime to n. For 15 = 3 · 5, ?(15) = 15·(1?1/3)·(1?1/5) = 15·(2/3)·(4/5) = 8. The formula for ? on a product of distinct primes applies here. Euler's Totient Function
How many positive divisors does 360 have?
18
20
24
30
360 = 2^3 · 3^2 · 5^1. The number of divisors is (3+1)(2+1)(1+1) = 4·3·2 = 24. Each exponent plus one multiplies for total divisor count. Divisor Function
What is the GCD of 2^5·3^3·7 and 2^3·3·5·7^2?
210
672
168
294
Take the minimum exponent for each common prime: 2^min(5,3)=2^3, 3^min(3,1)=3^1, 7^min(1,2)=7^1 gives 8·3·7 = 168. This method applies the fundamental GCD calculation via prime exponents.
What is the LCM of 2^3·3 and 2·3^2·5?
180
360
240
120
For LCM take the maximum exponent: 2^max(3,1)=2^3, 3^max(1,2)=3^2, and include 5^1 gives 8·9·5 = 360. LCM captures all prime requirements at their highest counts.
Factor x^3 - y^3.
(x - y)(x^2 + xy + y^2)
(x - y)^3
(x + y)(x^2 - xy + y^2)
(x - y)(x + y)^2
x^3 - y^3 is the difference of cubes: (a - b)(a^2 + ab + b^2) with a=x, b=y. This expands back to x^3 - y^3. Recognizing the cube formula is essential.
What is gcd(p^2 q^3, p^3 q^2 r^4) for distinct primes p, q, r?
p^4 q^2
p^2 q^2
p^3 q^3
p q^4
To find the GCD, take each prime to the minimum exponent present: p^min(2,3)=p^2 and q^min(3,2)=q^2. The prime r is not common to both, so it's excluded.
How many trailing zeros does 100! have in base 10?
25
24
26
22
Trailing zeros come from factors of 10 = 2·5. In 100!, the limiting factor is the number of 5s: ?100/5? + ?100/25? = 20 + 4 = 24. Each contributes one zero in base 10.
Factor x^4 - y^4.
(x^2 - 2xy + y^2)(x^2 + 2xy + y^2)
(x - y)^4
(x^2 - y^2)(x^2 + y^2)
(x - y)(x^3 + y^3)
x^4 - y^4 is a difference of squares twice: (x^2 - y^2)(x^2 + y^2), and further x^2 - y^2 = (x - y)(x + y). Recognizing nested squares simplifies factorization.
How many positive integers less than 1000 are relatively prime to 1000?
625
250
400
500
1000 = 2^3 · 5^3, so ?(1000) = 1000·(1?1/2)·(1?1/5) = 1000·1/2·4/5 = 400. Euler's totient gives the count of co-prime integers below n. Euler's Totient Function
What is the smallest integer n > 1 such that ?(n) = 16?
32
20
17
15
We test candidates: ?(17)=16 since 17 is prime, ?(p)=p?1. Smaller n like 15 give ?(15)=8, 16, 20, etc. The smallest >1 yielding 16 is 17.
Using the Extended Euclidean Algorithm, find integers x and y such that 240x + 46y = 2.
x = -47, y = 9
x = 9, y = -47
x = -9, y = 47
x = 47, y = -9
Applying the extended algorithm yields the Bézout coefficients: 2 = 240·(-9) + 46·47. Back-substitution from gcd steps confirms x = -9, y = 47. These satisfy the linear combination.
Factor x^4 + 4y^4 using Sophie Germain's identity.
(x^2 + 2xy + 2y^2)(x^2 - 2xy + 2y^2)
(x + 2y)^2(x - 2y)^2
(x^2 + 4y^2)^2
(x^2 + y^2)^2 + 2xy
Sophie Germain's identity states x^4 + 4y^4 = (x^2 + 2xy + 2y^2)(x^2 ? 2xy + 2y^2). This factorization shows a sum of fourth powers can split into quadratics.
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Study Outcomes

  1. Understand prime factorization -

    Break down integers into their prime factors confidently, a core skill for any number theory quiz.

  2. Apply greatest common factor techniques -

    Use the Euclidean algorithm to compute greatest common factors in complex pairs, essential for the greatest common factors test.

  3. Solve algebraic factorization problems -

    Factor quadratic and higher-degree polynomials to simplify expressions, preparing you for algebraic factorization practice.

  4. Identify number classifications -

    Distinguish between prime, composite, and special number sets, reinforcing your number theory fundamentals.

  5. Analyze LCM and GCF relationships -

    Determine least common multiples and greatest common factors to solve real-world and abstract problems in the number theory practice test.

  6. Evaluate and refine strategies -

    Assess your problem-solving approach with instant feedback, guiding further study and improvement in subsequent math quiz questions.

Cheat Sheet

  1. Prime Factorization Fundamentals -

    The Fundamental Theorem of Arithmetic states every integer >1 can be uniquely written as a product of primes (MIT OpenCourseWare). For example, 84 = 2² × 3 × 7; mastering this is key for any prime factorization quiz. Visualize a factor tree to break numbers systematically into prime branches.

  2. Euclidean Algorithm for GCF -

    The Euclidean Algorithm, dating back to Euclid's Elements, efficiently finds the greatest common factor (GCF) of two numbers (e.g., GCF(48, 18)=6). Repeatedly apply a mod b until the remainder is zero for a foolproof greatest common factors test method. This approach is both faster and more reliable than listing divisors.

  3. Key Algebraic Factorization Patterns -

    Recognize patterns: difference of squares a²−b²=(a−b)(a+b), perfect square trinomial a²+2ab+b²=(a+b)², and sum/difference of cubes (a³±b³)=(a±b)(a²∓ab+b²) (Khan Academy). Practice these in your algebraic factorization practice routines to speed up solving. Spotting these forms at a glance will boost your accuracy on math quiz questions.

  4. Mnemonic Tricks & Shortcuts -

    Use "Down the diagonal, up on the cross" for difference of squares and "SOAP" (Same, Opposite, Always Positive) to pick signs in quadratics (University of Cambridge tips). Such mnemonic devices make recalling rules in a number theory practice test a breeze. Jot them on flashcards and quiz yourself during quick study sessions.

  5. Effective Practice & Self-Testing -

    Schedule frequent short quizzes using the spacing effect (research from the Learning Scientists) to reinforce concepts in your number theory quiz prep. Mix question types - prime factorization quiz, greatest common factors test, and algebra challenges - to simulate real test conditions. Track accuracy and time per question to identify weak spots and build confidence.

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