A kosarad üres
Highest common factor (HCF),
lowest common multiple (LCM)
lowest common multiple (LCM)
Example 1
Let us simplify the fraction \latex{\frac{1,020}{1,224}}.
Solution
We could solve it by simplifying by one number at a time and then check what else the new numerator and denominator can be further simplified by. Let us find the greatest number that the expression can be simplified by.
Let us do the prime factorisation of the numbers.
1,020 = 22 × 3 × 5 × 17; 1,224 = 23 × 32 × 17.
\latex{ 1,020 }
\latex{ 510 }
\latex{ 255 }
\latex{ 51 }
\latex{ 17 }
\latex{ 1 }
\latex{ 1,224 }
\latex{ 612 }
\latex{ 306 }
\latex{ 153 }
\latex{ 51 }
\latex{ 17 }
\latex{ 1 }
\latex{ 2 }
\latex{ 2 }
\latex{ 5 }
\latex{ 3 }
\latex{ 17 }
\latex{ 2 }
\latex{ 2 }
\latex{ 2 }
\latex{ 3 }
\latex{ 3 }
\latex{ 17 }
We can see that because of the common prime factors the two numbers have common factors. The highest common factor can be found formed from the common prime factors: 22 × 3 × 17 = 204.
When simplifying by this:
\latex{\frac{1,020}{1,224}=\frac{5}{6}}.
DEFINITION: In the case of two positive integers the greatest of the common factors is called the highest common factor of the two numbers.
The notation of the highest common factor of \latex{a;\,b}: HCF of \latex{a} and \latex{b}.
For example, in the previous case, HCF of \latex{1,020} and \latex{1,224=204}.
It can be proved that every common factor is a factor of the highest common factor.
The highest common factor can be constituted from the prime factorisation form by multiplying the powers received when raising the common prime factors to the appearing smallest index.
Example 2
Let us find the highest common factor of the following number pairs:
- \latex{ 73,125} and \latex{7,425 };
- \latex{4,617} and \latex{6,800}.
Solution (a)
The prime factorisation of the numbers:
\latex{73,125=3^2\times5^4\times13}; \latex{7,425=3^3\times5^2\times11}.
Thus, the HCF of \latex{73,125} and \latex{7,425=3^2\times 5^2=225}.
Solution (b)
The prime factorisation of the numbers:
\latex{4,617=3^5\times19;\,\,6,800=2^4\times5^2\times17}.
There is no common factor in the two prime factorisation forms. It means that these numbers have only one common factor which is number 1.
Thus the HCF of \latex{4,617} and \latex{6,800=1}.
Thus the HCF of \latex{4,617} and \latex{6,800=1}.
DEFINITION: Positive integers whose highest common factor is \latex{1} are called relatively prime or coprime.
It is important to see that the fact that two distinct numbers are relatively prime does not necessarily mean that these are primes, but if these are primes then these are surely relatively prime too. For example, HCF of \latex{15} and \latex{8=1}; HCF of \latex{11} and \latex{43=1}; but HCF of \latex{11} and \latex{275=11}.
We can talk about the highest common factor of not only two numbers, but of three or more numbers too. For example, HCF of \latex{7,425;\,6,800} and \latex{73,125=5^2=25} based on the previous prime factorisations.
Example 3
Let us do the addition \latex{\frac{1}{1,176}+\frac{1}{720}}.
Solution
We could choose the product of the two denominators as the common denominator. However, simplifying large numbers might become too difficult, so let us try to find the smallest common denominator.
The prime factorisation of the denominators: \latex{1,176=2^3\times3\times7^2; \,\,720=2^4\times3^2\times5}.
If we multiply all the appearing prime factors (the common ones raised to the larger index), then we get a multiple of both numbers, and this multiple will be the smallest: \latex{2^4\times3^2\times5\times7^2=35,280}.
\latex{ 1,176 }
\latex{ 588 }
\latex{ 294 }
\latex{ 147 }
\latex{ 49 }
\latex{ 7 }
\latex{ 1 }
\latex{ 720 }
\latex{ 360 }
\latex{ 180 }
\latex{ 90 }
\latex{ 45 }
\latex{ 15 }
\latex{ 5 }
\latex{ 1 }
\latex{ 2 }
\latex{ 2 }
\latex{ 2 }
\latex{ 3 }
\latex{ 7 }
\latex{ 7 }
\latex{ 2 }
\latex{ 2 }
\latex{ 2 }
\latex{ 2 }
\latex{ 3 }
\latex{ 3 }
\latex{ 5 }
The addition: \latex{\frac{1}{2^3\times3\times7^2}+\frac{1}{2^4\times3^2\times5}=\frac{2\times3\times5+7^2}{2^4\times3^2\times5\times7^2}=\frac{79}{35,280}}
DEFINITION: In the case of two positive integers, the smallest positive number of the common multiples is called the lowest common multiple of the two numbers.
The notation of the lowest common multiple of \latex{ a } and \latex{b}: LCM of \latex{a} and \latex{b}.
For example, in the previous case: LCM of \latex{1,176} and \latex{720=35,280}.
The lowest common multiple can be deduced from the prime factorisation form by multiplying the powers received when raising all the appearing prime factors to the appearing largest index.
Example 4
Let us find the lowest common multiple of numbers \latex{ 972; \,8,775 }.
Solution
The prime factorisation of the numbers:
\latex{972=2^2\times3^5;\,\,8,775=3^3\times5^2\times13}.
The lowest common multiple is: LCM of \latex{972} and \latex{8,775=2^2\times3^5\times5^2\times13=315,900}.
⯁ ⯁ ⯁
We can realise that if two numbers are relatively prime then their lowest common multiple will be the product of the two numbers.
Exercises
{{exercise_number}}. Simplify the following fractions.
- \latex{\frac{1,425}{1,725}}
- \latex{\frac{3,168}{52,272}}
- \latex{\frac{39,375}{18,375}}
{{exercise_number}}. Along the road there are trees every \latex{ 24 } \latex{ metres } on one side and electric poles every \latex{ 51 } \latex{ metres } on the other side. There is a place where there is a tree and an electric pole opposite each other. After what distance does it happen again?
{{exercise_number}}. Two buses set off from a bus station, one of them sets off every \latex{ 30 } \latex{ minutes }, the other one every \latex{ 25 } \latex{ minutes }. At \latex{ 6 } AM, both buses set off. Until noon, how often does it happen that the buses set off at the same time?
{{exercise_number}}. Is it true that two adjacent positive integers are always relatively prime?
{{exercise_number}}. The sum of two positive integers is \latex{ 175 }, their highest common factor is \latex{ 35 }. Give these numbers.
{{exercise_number}}. Give three numbers which are relatively prime, but the highest common factor of any two of them is greater than \latex{ 1 }.
{{exercise_number}}. We know that \latex{a\mid b}. What are the LCM of \latex{a} and \latex{b} the HCF of \latex{a+b} and \latex{b} equal to?
{{exercise_number}}. For what numbers \latex{ a } is it true that the LCM of \latex{a} and \latex{24} is \latex{72?}
{{exercise_number}}. What is the smallest natural number divisible by \latex{ 7 } which leaves \latex{ 1 } as a remainder when divided by \latex{ 2;\, 3;\, 4;\, 5 } and \latex{ 6? }
{{exercise_number}}. Prove that if \latex{ a } and \latex{ b } are natural numbers, then its HCF\latex{\times}LCM\latex{=a\times b}.
Quiz
Give positive integers \latex{ a } and \latex{ b } and prime number \latex{ p } for which it is fulfilled that
LCM of \latex{a} and \latex{b\,+\,}HCF of \latex{a} and \latex{b=a+b+p}.
