Prime Numbers and Prime Factorization

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).

1. a) Why can they not do this with $13$ students in any way other than a single row of $13$?
2. 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 factors):

$11, 14, 19, 21, 23, 27, 31, 33$

Classify the following numbers as prime or composite, justifying your answer using the divisibility rules. For the composite numbers, state at least one divisor other than $1$ and the number itself:

$47, 51, 67, 87, 91, 101$

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 $1$ is not considered prime and why the number $9$ gets trapped in the filter for composite numbers.

In a math 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.

To check whether the number $601$ is prime, a student claims that they only need to divide it by the prime numbers less than $25$.

1. a) Justify why the student has chosen the number $25$ as a limit.
2. b) Carry out the necessary divisions following their reasoning and determine whether $601$ is prime or composite.
3. c) What is the largest prime number you would have to test to know if $1009$ is prime?

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:

1. a) $180$
2. b) $252$
3. c) $450$

Sometimes, to factorize numbers ending in zero, we can use mental "shortcuts". For example, we know that $10 = 2 \times 5$, 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 $2 \times 5$ on the right):

1. a) $1,200$
2. b) $4,500$
3. c) $18,000$

Fill in the blanks in this factor tree for the number $120$. Remember that each branch must end in a prime number for the factorisation to be complete. Then write the final result using powers.

Look at the following "division tower" (prime factorisation) and identify the missing numbers in the blank spaces so that it is correct:

The number $12$ can be represented in a rectangular shape in several ways (for example, $2 \times 6$ or $3 \times 4$). However, a prime number can only form a single row.

Look at the drawing and find the prime factorisation of the total number of dots forming the rectangle:

Find out which number it is in each case by working out the powers and multiplying the factors:

1. a) $2^2 \times 3 \times 5 = $
2. b) $2 \times 3^3 = $
3. c) $3^2 \times 11 = $

In the math lab, an incomplete prime factorization has been left on the board. Help reconstruct it knowing that the original number is less than $500$.

Calculate the value of the original number $X$, the intermediate quotient $Y$ and the prime factor $P$ knowing that $P$ is the smallest possible prime number that makes $X$ a multiple of $3$.

Find all the factors of the number $N$ knowing that its prime factorisation is $N = 2^2 \times 13$. 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:

1. a) $28$
2. b) $42$

Find the smallest number that has exactly $6$ factors, knowing that only the first two prime numbers appear in its prime factorisation. Explain your reasoning by analysing the exponents in its factorisation.

Marta has a collection of old coins. If she groups them $6$ by $6$, she has $3$ left over. If she groups them $10$ by $10$, she also has $3$ left over.

Knowing that she has more than $50$ coins but fewer than $80$, exactly how many coins does Marta have in her collection? (Hint: relate the problem to the concept of least common multiple).

Given the numbers $A = 2^4 \times 3^2 \times 5 \times 11$ and $B = 2^3 \times 3^4 \times 7$.

Without calculating the numerical value of $A$ or $B$, answer giving reasons:

1. a) Is the product $A \times B$ divisible by $55$?
2. b) Is the product $A \times B$ a multiple of $54$?
3. c) What is the prime factorisation of the number resulting from $M.C.D.(A, B)$?

The area of a rectangular garden is $252$ $m^2$. If we know that its side lengths (in meters) are natural numbers and that the longer side is a multiple of $3$ and of $7$, but not of $2$, use the prime factorization of the number $252$ to find all possible dimensions of the garden.

A natural number $N$ has the following property: its prime factorisation is of the form $2^a \times 3^b$. We know that if we multiply $N$ by $2$, its number of factors increases by $3$, and if we multiply it by $3$, its number of factors increases by $4$.

Calculate the value of the exponents $a$ and $b$, and find the number $N$.

A "perfect" number is one that is equal to the sum of all its proper divisors (all its divisors except itself). For example, $6$ is perfect because its proper divisors are $1, 2$ and $3$, and $1 + 2 + 3 = 6$.

1. a) Check whether the number $28$ is a perfect number.

1. b) The next perfect number is greater than $400$ and less than $500$. Knowing that it is a multiple of $16$ and of $31$, use prime factorization to find it.

In cybersecurity, very large prime numbers are used to create keys. Suppose a security key is generated by multiplying two distinct prime numbers, $p$ and $q$. The result is $N = p \times q$.

If we know that $N = 1591$ and that one of the prime numbers is close to the square root of $N$:

1. a) Find the values of $p$ and $q$.
2. b) How many total factors does the number $N$ have? Justify your answer without listing them.