您的購物車是空的
Common factors, the highest common factor
An \latex{ 8 } and a \latex{ 12 }-unit block is made using smaller elements. How long can the longest element be used to construct both blocks?
Example 1
At a workshop, children make figures using \latex{ 12 } pine cones and \latex{ 30 } chestnuts. How many children participate in the workshop if everyone gets the same number of pine cones and chestnuts?
Solution
Since \latex{ 12 } pine cones and \latex{ 30 } chestnuts must be divided equally, the number of children must be a factor of both \latex{ 12 } and \latex{ 30 }.
Factors of \latex{ 12 }:
Factors of \latex{ 30 }:
\latex{1}
\latex{1}
\latex{2}
\latex{2}
\latex{3}
\latex{3}
\latex{6}
\latex{6}
\latex{4}
\latex{12}
\latex{5}
\latex{10}
\latex{15}
\latex{30}
The blue numbers are the common factors of \latex{ 12 } and \latex{ 30 }.
pine cones
chestnuts
\latex{ 1 } child
\latex{ 2 } children
\latex{ 3 } children
\latex{ 6 } children
\latex{12}
\latex{6}
\latex{4}
\latex{2}
\latex{30}
\latex{15}
\latex{10}
\latex{5}
There might be \latex{ 1,\, 2,\, 3, } or \latex{ 6 } children participating in the workshop.
In the previous example, you had to find the numbers that are factors of both \latex{ 12 } and \latex{ 30 }. Such factors are called the common factors of \latex{ 12 } and \latex{ 30 }.
Their common factors are \latex{ 1,\, 2,\, 3, } and \latex{ 6 }. Therefore, their greatest common factor (or greatest common divisor) is \latex{ 6 }.
The greatest common factor is written as: \latex{ (12; 30) = 6 }.
Common factors of \latex{ 12 } and \latex{ 30 }
Factors of \latex{12}
Factors of \latex{30}
\latex{4}
\latex{12}
\latex{1}
\latex{2}
\latex{3}
\latex{6}
\latex{15}
\latex{5}
\latex{10}
\latex{30}
The common factors of two numbers are factors of both numbers. The largest of the common factors is the greatest common factor.
To determine the common factors of two natural numbers, you can use prime factor decomposition.
\latex{12}
\latex{6}
\latex{3}
\latex{1}
\latex{30}
\latex{15}
\latex{5}
\latex{1}
\latex{5}
\latex{2}
\latex{2}
\latex{3}
\latex{2}
\latex{3}
\latex{12=2\times}
\latex{2}
\latex{\times}
\latex{3}
\latex{30=}
\latex{\times}
\latex{2}
\latex{3}
\latex{\times}
\latex{5}
The common factors of \latex{ 12 } and \latex{ 30 } are:
\latex{1;2;3;2\times 3}
The greatest common factor: \latex{ (12; 30) = 2 × 3 = 6 }.
Example 2
For a toy sale, \latex{ 36 } stickers, \latex{ 60 } figures, and \latex{ 90 } marbles were collected. How many packages can be made at most if they must contain the same number of items?
Solution
The organisers want to make the most packages possible. Therefore, they must find the greatest common factor of \latex{ 36,\, 60, } and \latex{ 90 }. Using prime factor decomposition, you can determine all the factors of \latex{ 36,\, 60, } and \latex{ 90 }. Some of them are factors of all three numbers.
\latex{36}
\latex{18}
\latex{9}
\latex{3}
\latex{1}
\latex{60}
\latex{30}
\latex{15}
\latex{5}
\latex{1}
\latex{90}
\latex{45}
\latex{15}
\latex{5}
\latex{1}
\latex{2}
\latex{3}
\latex{2}
\latex{5}
\latex{5}
\latex{3}
\latex{2}
\latex{3}
\latex{2}
\latex{3}
\latex{2}
\latex{3}
\latex{36=}
\latex{60=}
\latex{90=}
\latex{2}
\latex{2}
\latex{2}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{2}
\latex{2}
\latex{3}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{3}
\latex{3}
\latex{3}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{3}
\latex{5}
\latex{5}
The greatest common factor of \latex{ 36,\, 60, } and \latex{ 90 } is
\latex{(36; 60; 90) = 6}.
Only \latex{ 6 } packages with the same content can be made.
The greatest common factor can be used to simplify fractions.
Example 3
Which number can be used to simplify the fraction \latex{ \frac {36}{90} } so that it cannot be further simplified?
Solution
Find the greatest common divisors of \latex{36} and \latex{90}.
\latex{36}
\latex{18}
\latex{9}
\latex{3}
\latex{1}
\latex{90}
\latex{45}
\latex{15}
\latex{5}
\latex{1}
\latex{2}
\latex{5}
\latex{3}
\latex{3}
\latex{2}
\latex{2}
\latex{3}
\latex{3}
\latex{36=2\times}
\latex{2}
\latex{\times}
\latex{3}
\latex{\times}
\latex{3}
\latex{90=}
\latex{2}
\latex{\times}
\latex{3}
\latex{\times}
\latex{3}
\latex{\times}
\latex{5}
\latex{(36;90)=2\times3\times3=18}
Reduce the fraction by \latex{18}.
\latex{\frac{36}{90}}
\latex{=}
\latex{\frac{2}{5}}
\latex{\div 18}
\latex{\div 18}
We can write it using prime product form:
\latex{\frac{36}{90} =\frac{2\times \cancel{2}\times \cancel{3}\times \cancel{3}}{\cancel{2}\times \cancel{3}\times \cancel{3}\times 5}=\frac{2}{5}}
The fraction \latex{\frac{2}{5}} cannot be reduced any further. We reduced it using the greatest common divisor.
\latex{\frac{36}{90}}
\latex{\frac{18}{45}}
\latex{\frac{36}{90}}
\latex{\frac{12}{30}}
\latex{\frac{36}{90}}
\latex{\frac{6}{15}}
\latex{\frac{36}{90}}
\latex{\frac{4}{10}}
\latex{=}
\latex{=}
\latex{=}
\latex{=}
\latex{\div 2}
\latex{\div 2}
\latex{\div 3}
\latex{\div 3}
\latex{\div 9}
\latex{\div 9}
\latex{\div 6}
\latex{\div 6}
The resulting fractions can be reduced further
Example 4
Reduce the fraction \latex{\frac{12}{35}}
Solution
Use the prime product form to find the greatest common divisor.
\latex{12}
\latex{6}
\latex{3}
\latex{1}
\latex{35}
\latex{7}
\latex{1}
\latex{2}
\latex{2}
\latex{3}
\latex{5}
\latex{7}
\latex{12=2\times 2\times 3} \latex{35=5\times7}
Common factors
Factors of \latex{12}
Factors of \latex{35}
\latex{4}
\latex{2}
\latex{3}
\latex{6}
\latex{12}
\latex{5}
\latex{7}
\latex{35}
\latex{1}
There are no common factors, the greatest common divisor is \latex{1}.
Since \latex{(12;35)=1}, thus \latex{\frac{12}{35}}
If two numbers do not have a common prime factor, then their greatest common divisor is \latex{1}. These numbers are called relative prime numbers.
For example: \latex{4} and \latex{9} are relative primes: \latex{(4; 9) = 1}, since their only common divisor is \latex{1}.
\latex{4}
\latex{2}
\latex{1}
\latex{2}
\latex{2}
\latex{9}
\latex{3}
\latex{3}
\latex{3}
\latex{1}
\latex{4=2\times2}
\latex{9=3\times3}
Since \latex{4} and \latex{9} are relative primes, a number is divisible by \latex{4\times9=36} if it is divisible by \latex{4} and by \latex{9}.
Exercises
{{exercise_number}}. Find the greatest common factor of the following numbers. Make a Venn diagram showing the factors.
- \latex{(72; 60)}
- \latex{(52; 64)}
- \latex{(126; 294)}
- \latex{(1,512; 1,872)}
- \latex{(48; 72)}
{{exercise_number}}. Gift packs were given to every first grader. The packs contained \latex{ 168 } notebooks, \latex{ 420 } pencils and \latex{ 252 } pens in total. How many first graders could receive a gift pack?
{{exercise_number}}. \latex{ 42 } boys and \latex{ 72 } girls participated in a competition for \latex{ 6 }th graders. The organisers want to form teams with the same number of students and the same ratio of boys and girls. How many teams can be formed at most?
{{exercise_number}}. Simplify the following fractions by the greatest common factor of the numerator and the denominator.
- \latex{\frac{72}{60}}
- \latex{\frac{52}{64}}
- \latex{\frac{48}{72}}
- \latex{\frac{126}{294}}
- \latex{\frac{36}{72}}
- \latex{\frac{16}{25}}
{{exercise_number}}. The pirates found a treasure chest on a desert island. The chest contained \latex{ 196 } silver coins, \latex{ 182 } gold coins and \latex{ 126 } pearls. How many pirates were on the island at most if the treasure was divided equally among them?
{{exercise_number}}. Write the greatest common factors of numbers A and B as the product of their prime factors.
- \latex{A=2\times 2\times 2\times 3\times 3\times 5\times 7\times 7\\B=2\times 2\times 3\times 3\times 5\times 5\times 7\\(A;B)=?}
- \latex{A=2\times 7\times 7\times 7\times 7\times 11\times 11\times 13\\B=2\times 2\times 2\times 2\times 7\times 7\times 7\times 11\times 19\\(A;B)=?}
- \latex{A=5\times 5\times 5\times 7\times 7\times 17\times 23\\B=5\times 5\times 5\times 5\times 7\times 23\times 29\\(A;B)=?}
{{exercise_number}}. Multiply the primes on number cards \latex{2} \latex{3} \latex{5} and \latex{7} to create all the possible composite numbers that you can create using these cards.
- How many composite numbers can you create?
- Which one is the largest?
- What is the greatest common factor of the smallest and the largest composite number?
- Are all the other composite numbers factors of the largest composite number?
{{exercise_number}}. Are there
- two even numbers whose greatest common factor is \latex{ 1 } (relative primes);
- two composite numbers whose greatest common factor is \latex{ 1 };
- two neighbouring natural numbers that have a common factor greater than \latex{ 1 }?
Justify your answers.
{{exercise_number}}. How many times should factor \latex{ 5 } appear among the prime factors of number \latex{ A }, and how many times should factor \latex{ 2 } appear among the prime factors of number \latex{ B } so that their greatest common factor would be \latex{2\times 2\times 2\times 5\times 5\times 7 } ?
\latex{(A;B)=2\times 2\times 2\times 5\times 5\times 7}
\latex{A=2\times 2\times 2\times 2\times \underbrace{5\times ...\times 5}_{\text{y}}\times 7\times 11}
\latex{B=\underbrace{2\times ...\times 2}_{\text{x}}\times 3\times 5\times 5\times 7}
{{exercise_number}}. The greatest common factor of two natural numbers is \latex{ 1 }, and their product is \latex{ 168 }. What are these two numbers?
{{exercise_number}}. Decide whether the following statements are true or false. Justify your answers.
- If the greatest common factor of two numbers is \latex{ 1 }, then they do not have a common prime factor.
- The greatest common factor contains the common prime factors of two numbers.
- If two numbers do not have common prime factors, they do not have common factors either.
- If you choose two random numbers, it is possible that one of the numbers is the greatest common factor of the two numbers.
Quiz
The area of a rectangle is \latex{ 126 } units. How long can its sides be if the lengths of the sides are coprimes?
