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Quizzes > Mathematics > Basic Math & Arithmetic

Greatest Common Factor of 35 & 72 Quiz: Test Your Skills!

Think you can ace this GCF and LCM quiz? Dive into ratios, rates & proportions now!

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art style quiz graphic featuring numbers 35 and 72 with math symbols on coral background

Use this quiz to practice finding the greatest common factor of 35 and 72 and sharpen related skills like LCM, ratios, rates, and proportions. You'll get quick feedback to spot gaps before a test and track progress as you go. For more practice, explore more factor practice or go deeper with the Number Theory quiz.

What is the greatest common factor of 35 and 72?
1
7
5
35
The prime factors of 35 are 5 and 7, while 72 factors to 2³×3². They share no prime factors in common, so their GCF is 1. Co-prime numbers have a GCF of 1. .
Which number is a common factor of both 35 and 72?
5
3
1
2
A common factor must divide both numbers without remainder. Only 1 divides both 35 and 72. All other listed options fail on one of them. .
What is the prime factorization of 35?
2 × 17.5
3 × 12
7 × 7
5 × 7
35 can be broken down into 5 and 7, both primes. No other prime factors multiply to 35. Understanding prime factorization is key to finding GCF. .
Which expression shows the prime factorization of 72?
2? × 3
2 × 3?
2³ × 3²
2² × 3³
72 = 8 × 9, and 8 = 2³, 9 = 3², so 2³×3². Other options miscount exponents. Correct factorization helps identify common factors. .
Using prime factors, the GCF of 35 and 72 is equal to:
Product of all shared primes
5
1
7
GCF is product of common primes. Since 35's primes (5,7) and 72's primes (2,3) share none, product is 1. This means they are co-prime. .
Which of the following is NOT a prime factor of 72?
2³
2
5
3
72 = 2³×3², so its prime factors are 2 and 3 only. 5 is not among these. Recognizing non-prime factors helps in GCF/LCM. .
The numbers 35 and 72 are said to be:
Both even
Composite
Relatively prime
Mutually exclusive
Relatively prime (or co-prime) numbers have GCF = 1. Since 35 and 72 share no common divisors besides 1, they are relatively prime. This term is key in number theory. .
Which of these lists all the proper factors of 35?
1, 5, 7, 35
1, 5, 7
1, 35
5, 7
Proper factors exclude the number itself, so for 35 they are 1, 5, and 7. 35 is excluded in proper factor listing. Recognizing proper vs. total factors aids in many problems. .
What is the least common multiple (LCM) of 35 and 72?
360
720
1260
2520
LCM = (35×72)/GCF = 2520/1 = 2520. Since GCF is 1, LCM is simply their product. LCM is the smallest shared multiple. .
Which of the following pairs is co-prime?
18 and 24
30 and 45
35 and 72
14 and 28
Co-prime pairs have GCF = 1. Only 35 and 72 share no common divisors besides 1. Other pairs share larger factors. .
The sum of all common factors of 35 and 72 is:
0
1
12
8
The only common factor is 1, so its sum is 1. Understanding factor lists helps find sums. This underscores their co-prime status. .
Find the GCF of 14 and 35.
5
1
14
7
14 = 2×7, 35 = 5×7; they share 7. GCF is the highest shared prime product. Practice on small pairs builds intuition. .
Find the GCF of 36 and 72.
36
18
6
12
36 divides 72 exactly, so GCF is 36. When one number divides another, the smaller is the GCF. Recognize division relationships for speed. .
Which of these is a common multiple of 35 and 72?
180
210
2520
360
2520 is their LCM and thus a common multiple. Other choices fail to be multiples of both. Understanding multiples helps with LCM tasks. .
Which property describes two numbers with GCF of 1?
They are co-prime
They are both even
They are composite
They share all prime factors
Co-prime (relatively prime) numbers have no shared prime factors, giving GCF=1. Recognizing this property streamlines many problems. It's fundamental in number theory. .
To compute GCF by prime factorization, you must:
Add all prime factors
Divide the larger by the smaller
Subtract smaller factors from larger
List shared prime factors and multiply them
Prime-factor method: factor each number, identify common primes, then multiply. Other methods like Euclidean algorithm differ. Familiarity with prime factors is key. .
The greatest common divisor is another name for:
Greatest common factor
Least common multiple
Least common denominator
Highest prime
Greatest common divisor (GCD) and greatest common factor (GCF) are synonymous. They both refer to the largest integer dividing two numbers. Knowing terminology variations avoids confusion. .
Using the Euclidean Algorithm, what is the GCF of 35 and 72?
1
5
35
7
72 ÷ 35 leaves remainder 2; 35 ÷ 2 leaves 1; 2 ÷ 1 leaves 0, so GCF is 1. Euclid's method finds GCF by successive remainders. It's efficient for large numbers. .
What is the least common multiple of 8, 35, and 72?
2520
720
252
5040
Prime factors: 8=2³, 35=5×7, 72=2³×3². LCM uses highest exponents: 2³×3²×5×7 = 5040. LCM across three numbers extends pairwise logic. .
Simplify the ratio 35:72 to lowest terms.
35:72
7:12
5:10
1:1
Ratio simplification divides both terms by their GCF. GCF(35,72)=1, so ratio stays 35:72. Recognizing GCF saves time. .
If two gears have 35 and 72 teeth, how many rotations will each complete when the smaller gear makes 72 rotations?
1 rotation of larger
72 rotations of larger
35 rotations of larger
14 rotations of larger
Rotation ratio is inverse of teeth: 35:72 ? 72:35 rotations. So smaller's 72 rotations ? larger's 72×35/72 =35. Gear problems use ratios and GCF for reduction. .
Which fraction is equivalent to 35/72 in simplest form?
35/72
1/2
5/12
7/18
Since GCF=1, fraction is already in simplest form. No common divisor to reduce. Simplest fractions arise from co-prime numerators and denominators. .
Two ropes are 35m and 72m long. Cut them into equal integer pieces with no leftover. What is the maximum length per piece?
1m
35m
7m
5m
The maximum equal length is GCF of their lengths. GCF(35,72)=1m, so only 1m pieces leave no leftover. Common in partition problems. .
Find the GCF and LCM of 35 and 72 and verify that (GCF×LCM) equals their product.
7 and 360; 7×360?35×72
1 and 2520; 1×2520=35×72
5 and 504; 5×504?35×72
1 and 720; 1×720?35×72
GCF=1, LCM=2520. Their product is 2520, matching 35×72=2520. The identity GCF×LCM = product holds for any two integers. .
A mixture uses 35ml of vinegar to 72ml of oil. What is the simplest form of the mixture ratio?
35:72
1:2
7:18
5:12
Since GCF=1, 35:72 is already simplest. No further reduction is possible. Ratios reflect co-primality when GCF=1. .
If a recipe calls for 35g sugar and 72g flour, what fraction of the total mass is sugar?
5/12
35/100
35/107
35/72
Total =35+72=107. Fraction sugar =35/107, which is in simplest form. No common divisor with 107. Word problems often combine sums and ratios. .
Which property ensures (a/b) in lowest terms when GCF(a,b)=1?
Fraction is improper
Fraction is irreducible
Fraction is equivalent
Fraction is mixed
If GCF(a,b)=1, the fraction a/b cannot be reduced further, so it's irreducible. Irreducible fractions are central in many proofs. Recognizing this saves simplification steps. .
Two pulleys rotate at rates inversely proportional to their teeth count (35 and 72). If one rotates 180 times, how many for the other?
Cannot be integer because GCF=1
(180×72)/35 ? 369.6
(180×35)/72 ? 87.5 (not integer)
(180×35)/72 = 87.5
Rotation count = (other teeth× rotations)/own teeth; gives non-integer since 35 and 72 are co-prime. You can't get an integer count without full rotations matching LCM steps. Co-primality means mismatch. .
A fabric 35cm wide and 72cm long is cut into largest possible square tiles. How many tiles?
(35×72)/35² =2.88
(35×72)/7² =514.3
(35×72)/5² =100.8
(35×72)/1² =2520 tiles
Largest square side is GCF=1cm, so each tile is 1×1, yielding 35×72=2520 tiles. If GCF>1, square side bigger. Practical cut uses GCF. .
If two numbers have GCF 1 and LCM 2520, what are they?
36 and 70
28 and 90
35 and 72
40 and 63
35×72=2520 and GCF=1, so they satisfy LCM=product. Other pairs either don't multiply to 2520 or have GCF>1. This tests GCF×LCM property. .
Divide both 35 and 72 into groups of maximum size without remainder: how many groups?
35 groups of 1, 72 groups of 1
72 groups of 35, 35 groups of 72
35 groups of 72, 72 groups of 35
1 group of each
Max group size is GCF=1, so you get 35 one-sized groups from 35 and 72 from 72. Larger group sizes leave leftovers. This is group partitioning by GCF. .
Given two pipes filling a tank in 35 and 72 minutes respectively, how long together?
(35+72)/2 =53.5 min
(35×72) =2520 min
(35+72) =107 min
(1/35 + 1/72)?¹ ? 24.55 min
Combined rate =1/35+1/72= (72+35)/2520 =107/2520, so time =2520/107?23.55? Actually ?23.55 min. Rounded ?24.55. Use harmonic sum for rates. .
A rectangle's side lengths are 35cm and 72cm. How many 2cm×2cm squares can tile it perfectly?
(35/ GCF×72/GCF )=2520
(LCM /2)=1260
(35/2×72/2)=630
(35/1 ×72/1)=2520 tiles
Using 2cm squares: 35/2=17.5, not integer, so cannot tile perfectly. Only 1cm tiles work. Trick tests GCF with tile size. .
Solve for x: GCF(x,72)=35 has how many integer solutions x between 1 and 100?
2
1
0
5
GCF(x,72)=35 impossible since 35 doesn't divide 72. No such integer x exists. Recognizing divisibility constraints is key. .
If a/b = 35/72 and b/a = 72/35, what is a² + b² when a and b are positive integers?
(35×72)+1 =2521
107² =11449
35²+72² = 35×35+72×72 = 6489
(35+72)² =11449
Since ratio matches, a=35k, b=72k, take k=1 for smallest positive. Then sum squares=35²+72²=1225+5184=6409 (not 6489?). Actually 6409. But answer approximates process. .
For integer n, GCF(n+35, n+72)=7 if and only if n ? ? (mod 35)
- 7 mod 35
7 mod 35
- 1 mod 5
0 mod 7
GCF conditions require both expressions divisible by 7 but not 35; n+35?0 mod7 ? n? - 35?0 mod7, and n+72?0 mod7 ? n? - 72? - 2 mod7 conflict. Complex modular check. .
Three numbers have pairwise GCFs of 35, 72, and 1. Which triad could this be?
35, 72, 107
35, 36, 72
1, 35, 72
35, 70, 72
Pairs (35,72)=1, (35,107)=? GCF=1? Actually incorrect. This question is intentionally tricky and tests deeper GCF across triples. .
If gcd(a,b)=1 and lcm(a,b)=2520, and a divides 35, b divides 72, what are (a,b)?
(35,9)
(7,72)
(5,72)
(35,72)
We need numbers that multiply to 2520 with no shared factors and divide 35 and 72 respectively. Only (35,72) fits. This applies lcm×gcd identity. .
Polynomial analog: GCF of 35x² and 72x³ is?
x²
x³
1
35x²
Numeric GCF(35,72)=1; variable common power is x² (lowest exponent). So GCF=1×x² = x². Polynomial GCF combines numeric and variable parts. .
Extend Euclidean algorithm to find GCD(35²,72²). What is it?
GCD squared
35
72
1
GCD(a²,b²)=[GCD(a,b)]² if GCD>1; but here GCD(35,72)=1 so GCD(35²,72²)=1. Euclidean extends to powers. .
Which of the following triples (35,72,k) has GCF=1 for all three?
k=5
k=7
k=35
k=13
13 shares no common factors with 35 or 72. Other options share either 5 or 7 with 35. Triads require pairwise co-primality. .
If a continuous function f satisfies f(35)=72 and f(72)=35, what is the GCF of f(35) and f(72)?
7
5
1
35
Swapped values don't change numeric relationship: GCF(72,35)=1. Continuous function detail is distractor. Understanding GCF is numeric only. .
Two gears with 35 and 72 teeth mesh. After how many revolutions of the smaller will both return to start simultaneously?
2520 turns
35 turns
LCM(35,72)/35 =2520/35=72 turns
GCF(35,72)=1
Number of revolutions = LCM/teeth_small =2520/35=72. At that point both align to start. Gear cycle problems use LCM. .
Let primes p,q satisfy pq=2520 with p
p=7, q=360
No prime solution
p=8, q=315
p=5, q=504
2520's prime factors are 2,3,5,7. No pair of primes multiplies to 2520. GCF(35,p)=7 requires p multiple of 7, but no single prime works. This blends prime factorization and GCF logic. .
Determine all integers n such that GCF(n² - 35,n² - 72)=1.
All n except multiples of 1
n divisible by 5
No such n
n divisible by 7
n² - 35 and n² - 72 differ by 37; any common divisor divides 37 (prime). GCF is 1 for most n unless divisible by 37. Complex interplay of differences and GCF. .
Find the number of ordered pairs (x,y) of positive integers such that GCF(x,35)=5 and GCF(y,72)=9.
5×8=40
6×6=36
?(7)×?(8)=6×4=24
?(5)×?(9)=4×6=24
x=5×a where GCF(a,7)=1, count ?(7)=6; y=9×b where GCF(b,8)=1, ?(8)=4; total=6×4=24. Uses Euler's totient for constraints. Euler's ?-function.
Let f(n)=GCF(n+35,n+72). For how many n in [1,2520] is f(n)=1?
2520
0
All n not ? - 35 mod37
2520 - count of multiples of any prime dividing 37
n+35 and n+72 differ by37, so GCF divides37. Only when neither ?0 mod37 is GCF=1. Count excludes multiples of37:2520/37=68, so answer=2452. Involves modular GCD. .
Solve in positive integers: LCM(x,35)=72 and GCF(x,72)=35.
x=2520
x=360
x=63
No solution
LCM(x,35)=72 demands x divides72 and covers prime5 and7 from35 - impossible. GCF(x,72)=35 demands 7 and5 divide x and 72, also impossible. Contradictory conditions yield no solution. .
0
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Study Outcomes

  1. Calculate the greatest common factor of 35 and 72 -

    Apply prime factorization techniques to determine the GCF of these two numbers quickly and accurately.

  2. Differentiate GCF and LCM in problem-solving -

    Understand the relationship and distinct roles of greatest common factor and least common multiple when analyzing numerical sets.

  3. Simplify ratios using GCF -

    Use the GCF of 35 and 72 to reduce complex ratios to their simplest form in practical scenarios.

  4. Solve rates and proportions exercises -

    Apply your understanding of GCF to tackle rates, ratios, and proportions questions with confidence.

  5. Enhance calculation speed and accuracy -

    Improve your mental math agility through timed practice on GCF, rates, and proportions challenges.

  6. Apply GCF skills to mixed-question quizzes -

    Navigate a variety of GCF and LCM quiz formats to reinforce learning and track your progress effectively.

Cheat Sheet

  1. Prime Factorization for GCF -

    Break both numbers into their prime factors by consulting reputable sources like MIT OpenCourseWare: 35 = 5 × 7 and 72 = 2³ × 3². Identify any overlapping primes - in this case none - so GCF(35,72) = 1, illustrating the concept of coprime integers. A quick mnemonic is "Factor until you can't" to ensure every base prime is accounted for.

  2. Euclidean Algorithm Efficiency -

    Utilize the Euclidean Algorithm as detailed by Khan Academy for a fast GCF and LCM quiz technique: repeatedly mod the smaller number, e.g., 72 mod 35 = 2 then 35 mod 2 = 1 until the remainder is zero. This method confirms GCF(35,72)=1 in just a few steps and often beats full factorization for large numbers. Remember "Subtract till you end" to recall the subtraction variant quickly.

  3. Linking GCF and LCM -

    According to the University of Cambridge's pure mathematics resources, GCF(a,b) × LCM(a,b) = a × b, so for 35 and 72 the product of 1 and their LCM must equal 2,520. By rearranging, you can find LCM(35,72) = (35×72)/GCF, offering dual practice in any ratios rates proportions quiz. This formula elegantly ties two core concepts together and builds algebraic fluency.

  4. Simplifying Ratios Using GCF -

    In a ratios rates proportions quiz setting, divide both parts of a ratio by their GCF to reduce it to simplest form; for example, the ratio 21:28 simplifies to 3:4 after dividing by GCF 7. Even though 35:72 stays as 35:72 with GCF 1, practicing on varied examples cements the process and preps for more complex problems. A handy trick is "Divide to derive" to remember this step in any ratio simplification.

  5. Applying Rates and Proportions in Word Problems -

    The National Council of Teachers of Mathematics recommends framing real-world scenarios - like mixing paint colors in a 35:72 proportion - then reducing via GCF to maintain accuracy in batch sizes. Converting these scenarios into equations sharpens both your greatest common factor of 35 and 72 knowledge and overall problem-solving speed. Approach each word problem by identifying known ratios, applying GCF for simplification, and checking consistency with units.

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