Prime numbers and prime factorisation
$13$ friends cannot line up in equal rows of more than one person: they only fit in a single row of $13$, or each on their own. In contrast, $12$ can be arranged in lots of ways: $2 \cdot 6$, $3 \cdot 4$, $1 \cdot 12$... The difference lies in their divisors.
Prime and composite numbers
In PE class, the teacher wants to arrange the $13$ students into perfect rows and columns for an exercise (for example, $2$ rows of $6$ people). a) Why can they not do…
In PE class, the teacher wants to arrange the $13$ students into perfect rows and columns for an exercise (for example, $2$ rows of $6$ people).
- a) Why can they not do this with $13$ students in any way other than a single row of $13$?
- b) What type of number is $13$ based on its factors?
Classify the following numbers in the table according to whether they are prime (they only have two factors: $1$ and themselves) or composite (they have more than two…
Classify the following numbers in the table according to whether they are prime (they only have two factors: $1$ and themselves) or composite (they have more than two factors):
$11, 14, 19, 21, 23, 27, 31, 33$
| Prime Numbers | Composite Numbers |
|---|---|
The Sieve of Eratosthenes
More than $2.000$ years ago, Eratosthenes came up with a "filter" to find all the primes up to a given number: you cross out the multiples of each prime you find, and whatever survives is prime.
The sieve as an algorithm
An algorithm is a precise list of instructions. Follow this version to find the primes up to 30:
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Write the numbers from 2 to 30. The 1 is left out because it is not prime.
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Circle the smallest number that is not yet circled or crossed out.
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Cross out its multiples that are greater than it, without crossing out the circled number itself.
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Repeat steps 2 and 3 until every number is either circled or crossed out. The circled numbers are the primes.
At a card collectors' convention, the display stands are numbered from $10$ to $25$. The organisers want to highlight stands with a prime number using an orange sign.…
At a card collectors' convention, the display stands are numbered from $10$ to $25$. The organisers want to highlight stands with a prime number using an orange sign. Write the list of all stand numbers that should have an orange sign.
The Sieve of Eratosthenes works like a filter that lets only prime numbers through. If we pour the numbers from $1$ to $10$ into the filter, explain why the number $1$…
The Sieve of Eratosthenes works like a filter that lets only prime numbers through. If we pour the numbers from $1$ to $10$ into the filter, explain why the number $1$ is not considered prime and why the number $9$ gets trapped in the filter for composite numbers.
In a maths class, the teacher says that to find out if a number less than $100$ is prime, it is sufficient to check that it is not divisible by $2, 3, 5$ and $7$.…
In a maths class, the teacher says that to find out if a number less than $100$ is prime, it is sufficient to check that it is not divisible by $2, 3, 5$ and $7$.
Explain why there is no need to check if it is divisible by $11$ or by other larger prime numbers. Apply this idea to check if the number $97$ is prime.
Prime factorisation
Every composite number can be written as a product of prime numbers: it is like finding the "building blocks" it is made of.
Find the prime factorisation of the number $40$ using the vertical line method and express the result as a product of powers.
Find the prime factorisation of the number $40$ using the vertical line method and express the result as a product of powers.
Find the prime factorisation of the following numbers using the vertical line method. Express the final result using powers where possible: a) $180$ b) $252$ c) $450$
Find the prime factorisation of the following numbers using the vertical line method. Express the final result using powers where possible:
- a) $180$
- b) $252$
- c) $450$
Sometimes, to factorize numbers ending in zero, we can use mental "shortcuts". For example, we know that $10 = 2 \cdot 5$, so each trailing zero contributes a factor $2$…
Sometimes, to factorize numbers ending in zero, we can use mental "shortcuts". For example, we know that $10 = 2 \cdot 5$, so each trailing zero contributes a factor $2$ and a factor $5$.
Factorize the following numbers using this trick to save steps in the vertical line (you can remove each trailing zero at once by writing $2 \cdot 5$ on the right):
- a) $1.200$
- b) $4.500$
- c) $18.000$
Divisors from the factorisation
Once you have the factorisation, you can find all the divisors by combining the factors, without having to test them one by one.
Find all the factors of the number $N$ knowing that its prime factorisation is $N = 2^2 \cdot 13$. Is $N$ a composite or prime number? Justify your answer based on the…
Find all the factors of the number $N$ knowing that its prime factorisation is $N = 2^2 \cdot 13$. Is $N$ a composite or prime number? Justify your answer based on the definition of these types of numbers.
Find all the factors of the following numbers using their prime factorisation. Show the process of combining factors: a) $28$ b) $42$
Find all the factors of the following numbers using their prime factorisation. Show the process of combining factors:
- a) $28$
- b) $42$
Back to the start: The group of $13$ friends can only form one row because $13$ is prime (its only divisors are $1$ and $13$); in contrast, $12$ is composite and has $6$ different divisors ($1, 2, 3, 4, 6$ and $12$), which allows many different arrangements.