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Primes and composite numbers
\latex{ 13 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 9 }
\latex{ 5 }
\latex{ 7 }
\latex{ 11 }
\latex{ 14 }
\latex{ 4 }
\latex{ 6 }
\latex{ 8 }
\latex{ 10 }
\latex{ 12 }
Classification of natural numbers according to the number of their factors
Example 1
List the factors of \latex{ 18; 24; 25; 29; 37; 2; 1; 0. }
Which numbers have more than two factors?
Which numbers have more than two factors?
Solution
The factors of \latex{ 18: \,1;} \latex{ 2;\, 3;\, 6;\, 9;} \latex{ 18. }
The factors of \latex{ 24: \,1;}\latex{\,2; \,3; \,4; \,6; \,8; \,12;} \latex{ 24. }
The factors of \latex{ 25: \,1;}\latex{ \,5;}\latex{\, 25. }
The factors of \latex{ 29: \,1; \,29. }
The factors of \latex{ 37: \,1; \,37. }
The factors of \latex{ 2: \,1; \,2. }
The factors of \latex{ 1: \,1. }
The factors of \latex{ 0: \,0; \,1; }\latex{\,2; \,3; \,4; \,5; \,6; \,7;} \latex{ \dots }
The factors of \latex{ 24: \,1;}\latex{\,2; \,3; \,4; \,6; \,8; \,12;} \latex{ 24. }
The factors of \latex{ 25: \,1;}\latex{ \,5;}\latex{\, 25. }
The factors of \latex{ 29: \,1; \,29. }
The factors of \latex{ 37: \,1; \,37. }
The factors of \latex{ 2: \,1; \,2. }
The factors of \latex{ 1: \,1. }
The factors of \latex{ 0: \,0; \,1; }\latex{\,2; \,3; \,4; \,5; \,6; \,7;} \latex{ \dots }
\latex{ 18; 24 } and \latex{ 25 } have more than \latex{ 2 } factors, while \latex{ 0 } has an infinite number of factors.
Composite numbers are natural numbers greater than \latex{ 1 } that have more than two positive factors.
There are numbers that have exactly two factors, for example, \latex{ 2; 29; 37 }. These are called prime numbers.
Prime numbers are natural numbers with exactly two positive factors.
The following are neither prime nor composite numbers:
- \latex{ 1 } - because it has exactly one positive factor,
- \latex{ 0 } - because it has an infinite number of factors (including every natural number).
Prime numbers from 1 to 100
There are several methods for finding prime numbers. The oldest one is called the sieve of Eratosthenes.
The process is as follows:
- List the numbers from \latex{ 1 } to \latex{ 100 }.
- Filter out (cross out) the composite numbers following these steps:
- Mark \latex{ 1 } with a square, since it is neither a composite number nor a prime. ▯
- \latex{ 2 } is not filtered out, as it is a prime. Mark it with a circle. Filter out every following number divisible by \latex{ 2 } (including the multiples of \latex{ 4; 6; 8 } and \latex{ 10 }) by crossing them out. ━
- \latex{ 3 } is a prime, as it is not a multiple of \latex{ 2 } and was not filtered out. Mark it with a circle. Filter out every following number divisible by \latex{ 3 } (including the multiples of \latex{ 6 }; \latex{ 9 }).︱
- \latex{ 5 } is a prime number, as it is not a multiple of \latex{ 2 } or \latex{ 3 }. Filter out the following numbers divisible by \latex{ 5 }. /
- \latex{ 7 } is not a multiple of any of the previous prime numbers, meaning that it is a prime number. Filter out the following numbers divisible by \latex{ 7 }. \
Since \latex{100 = 10 \times}\latex{10}, if any number smaller than \latex{ 100 } is expressed as a multiplication with two factors, one of the factors needs to be less than \latex{ 10 }.
The factors greater than \latex{ 10 } are not relevant, since the multiples of \latex{ 2; 3; 4; 5; 6; 7; 8 } and \latex{ 9 } have been filtered out.
Using this method, every composite number below \latex{ 100 } has been filtered out.
The sieve of Eratosthenes
Prime numbers from \latex{ 1 } to \latex{ 100 }
We are left with the prime numbers smaller than \latex{ 100 }:
\latex{2;3;5;7;11;13;17;19;23;29;31;37;41;43;47;53;59;61;67;71;73;79;83 } and \latex{97 }.
This method can also be applied to numbers greater than \latex{ 100 }.
Example 2
How many factors do the following numbers have?
Are they prime or composite numbers?
Are they prime or composite numbers?
- \latex{ 72 }
- \latex{ 16 }
- \latex{ 71 }
Solution
List the factors of the numbers using their factor pairs.
a)
\latex{ 2 }
\latex{ 7 }
\latex{ 1 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 6 }
\latex{ 8 }
\latex{ 9 }
\latex{ 2 }
\latex{ 8 }
\latex{ 4 }
\latex{ 6 }
\latex{ 2 }
\latex{ 7 }
\latex{ 3 }
\latex{ 2 }
\latex{ 1 }
\latex{ 6 } factor pairs, \latex{ 12 } factors
composite number
composite number
b)
\latex{ 6}
\latex{ 1 }
\latex{ 6 }
\latex{ 1 }
\latex{ 1 }
\latex{ 2}
\latex{ 4}
\latex{ 4}
\latex{ 8}
\latex{ 3 } factor pairs, \latex{ 5 } factors
composite number
composite number
c)
\latex{ 7 }
\latex{ 1 }
\latex{ 1 }
\latex{ 1 }
\latex{ 7 }
\latex{ 1 } factor pair, \latex{ 2 } factors
prime number
prime number
\latex{ 72 } has \latex{ 12 } factors, while \latex{ 16 } has \latex{ 5 } factors, meaning that they are composite numbers. \latex{ 71 } only has \latex{ 2 } factors, so it is a prime number.
Some composite numbers have an even number of factors. For example, \latex{ 72 } has \latex{ 12 } factors (its factor pairs consist of different numbers).
However, square numbers have an odd number of factors. For example, \latex{ 16 } has \latex{ 5 } factors (one of its factor pairs consists of the same numbers).
Interesting facts from the history of number theory
- Eratosthenes (\latex{ 276 } BC–\latex{ 194 } BC), the mathematician who developed the method for identifying prime numbers, was a true polymath. One of his many achievements was developing a method to calculate the radius of the Earth. The accuracy of his result is considered impressive even today. From \latex{ 240 } BC until his death, he was a librarian in the famous Library of Alexandria.
- Euclid (\latex{ 325 }? BC–\latex{ 265 } BC) proved that there were infinitely many prime numbers.
- While studying the prime and composite numbers, the Pythagoreans found numbers with unique properties.
Twin primes are primes whose difference is \latex{ 2 }. For example, \latex{ 11 } and \latex{ 13 } or \latex{ 17 } and \latex{ 19 }, etc. There is only one prime triplet: \latex{ 3; 5 } and \latex{ 7 }.
A Perfect number is a number equal to the sum of its positive factors excluding the number itself.
\latex{1 + 2 + 3 = 6}, and \latex{ 1; 2 } and \latex{ 3 } are the positive factors of \latex{ 6 } (excluding itself).
\latex{ 28 } is also a perfect number, since \latex{1 + 2 + 4 + 7 + 14 = 28}, and \latex{ 1; 2; 4; 7 } and \latex{ 14 } are the positive factors of \latex{ 28 } (excluding itself).
Two numbers are amicable numbers if they are related in a way that the sum of the factors of each number (excluding the number itself) is equal to the other number. For example, \latex{ 220 } and \latex{ 284 } are amicable numbers, since
the sum of the factors of \latex{ 220 } excluding itself: \latex{1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284};
the sum of the factors of \latex{ 284 } excluding itself: \latex{1 + 2 + 4 + 71 + 142 = 220}.
the sum of the factors of \latex{ 284 } excluding itself: \latex{1 + 2 + 4 + 71 + 142 = 220}.
Prime numbers have an important role even today, for example, in computer encryption and encoding.
Exercises
{{exercise_number}}. Find composite numbers with an odd number of positive factors.
{{exercise_number}}. How many positive integers are there that are smaller than \latex{ 400 } and have an odd number of positive factors? List them in ascending order. What can you determine about these numbers?
{{exercise_number}}. Calculate the sum of the factors of \latex{ 496 } excluding itself. What can you observe?
{{exercise_number}}. Solve the exercise using the following number cards.
\latex{ 2 }
\latex{ 3 }
\latex{ 5 }
\latex{ 3 }
\latex{ 3 }
\latex{ 5 }
\latex{ 3 }
\latex{ 5 }
\latex{ 7 }
Create three-digit numbers using all three cards. Answer the following questions in all three cases.
Of the listed three-digit numbers,
- how many are even;
- how many are divisible by \latex{ 3 };
- which are prime numbers;
- which are composite numbers?
{{exercise_number}}. Find all prime numbers greater than \latex{ 100 } and smaller than \latex{ 150 }.
{{exercise_number}}. Are the following numbers odd or even?
- the sum of the first \latex{ 25 } prime numbers
- the product of the first \latex{ 25 } prime numbers
- the sum of the first \latex{ 26 } prime numbers
- the product of the first \latex{ 26 } prime numbers
{{exercise_number}}. Based on the graph, represent natural numbers on the \latex{ x } - axis and their factors on the \latex{ y } - axis. (→)
\latex{ x }
\latex{ y }
\latex{ 11 }
\latex{ 10 }
\latex{ 9 }
\latex{ 8 }
\latex{ 7 }
\latex{ 6 }
\latex{ 5 }
\latex{ 4 }
\latex{ 3 }
\latex{ 2 }
\latex{ 1 }
\latex{ 0 }
\latex{ 11 }
\latex{ 10 }
\latex{ 9 }
\latex{ 8 }
\latex{ 7 }
\latex{ 6 }
\latex{ 5 }
\latex{ 4 }
\latex{ 3 }
\latex{ 2 }
\latex{ 1 }
\latex{ 12 }
{{exercise_number}}. Choose three of the following number cards so that
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ 5 }
\latex{ 6 }
\latex{ 7 }
\latex{ 8 }
\latex{ 9 }
- every three-digit number that can be created from them is a composite number;
- at least two of the three-digit numbers that can be created from them are prime numbers.
{{exercise_number}}. Twin primes are primes whose difference is \latex{ 2 }. Find twin primes between \latex{ 100 } and \latex{ 200 }.
Quiz
There are number groups consisting of three primes in which the sum of the two smaller primes is equal to the third prime. Which number appears in every such group?