Exercises: Prime numbers and prime factorisation

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 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):

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

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

Find the prime factorisation of the number $40$ using the vertical line method and express the result as a product of powers.

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

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

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:

Sofía wants to find out if the number $91$ is prime. To check it, she applies the most common divisibility tests: she notices that it is not even (so it is not divisible by $2$), that its digits add up to $9 + 1 = 10$, which is not a multiple of $3$ (so it is not divisible by $3$), and that it does not end in either $0$ or $5$ (so it is not divisible by $5$). With these data, Sofía concludes in her notebook: "Since it is not divisible by 2, or by 3, or by 5, 91 is a prime number."

1. a) Locate and explain the reasoning error that Sofía made according to the rule for determining whether a number is prime.
2. b) State what check Sofía forgot to carry out before giving her answer and justify why it was essential to do it.
3. c) Correct the exercise: determine whether the number $91$ is prime or composite and, if it is composite, write its prime factorisation.

You want to check whether the number $173$ is prime or composite without performing unnecessary divisions.

1. a) Estimate the value of $\sqrt{173}$ by indicating which two consecutive integers it lies between (hint: think of the closest perfect squares to $173$).
2. b) From your estimate, write the complete, ordered list of prime numbers by which it is strictly necessary to test divisibility. What is the maximum number of divisions you will have to do?
3. c) Perform the necessary divisions and determine whether $173$ is prime or composite. Justify why you can stop the process without needing to test larger prime numbers.

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:

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

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) $450$

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.

Find all the factors of the following numbers using their prime factorisation. Show the process of combining factors:

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

Nerea 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 Nerea have in her collection? (Hint: relate the problem to the concept of lowest common multiple).

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.

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):

1. a) $1.200$
2. b) $4.500$
3. c) $18.000$

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.

Two classmates, Hugo and Valeria, are arguing about the fastest and most reliable way to find out if the number $221$ is prime or composite:

• Hugo's method: he suggests checking one by one all the odd numbers from $3$ up to $110$ (roughly half of $221$).

• Valeria's method: she calculates mentally that $\sqrt{221} \approx 14{,}8$ and claims that it is only necessary to test dividing $221$ by prime numbers less than or equal to $14$ (that is: $2, 3, 5, 7, 11$, and $13$).

1. a) Compare both methods: how many divisions would Hugo have to perform compared to those Valeria needs? Explain why Valeria is right to discard composite numbers such as $9$.
2. b) Explain the mathematical basis of Valeria's method: why is it impossible for $221$ to have a divisor greater than $14$ if no divisor smaller than $14$ is found?
3. c) Apply Valeria's method to classify the number $221$ as prime or composite.

In a cross-country race of the School Sports Games of an autonomous community, $119$ athletes of the under-14 category are participating. The organisers want to divide the athletes into equal qualifying heats, such that each heat has the same number of runners, with more than $1$ runner per heat and without there being a single heat with all participants.

The head referee asserts that it is impossible to make equal groups because "the number 119 is not even and does not end in 0 or 5, so it is surely prime."

1. a) Model the situation using divisibility: translate the referee's claim and the organisers' requirement into a condition on the divisors of $119$.
2. b) Apply the prime-testing rule (bounding $\sqrt{119}$) to prove whether the referee is right or wrong.
3. c) If it is possible to organise the heats, list all the different ways the athletes can be grouped, specifying the number of heats and the number of runners per heat.

A natural number $N$ has the following property: its prime factorisation is of the form $2^a \cdot 3^b$. 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 $N$.

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?

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 $N = p \cdot q$.

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.

In the maths lab, an incomplete prime factorisation 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 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.

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

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

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

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

We are looking for an unknown natural number $N$ that simultaneously satisfies three conditions:

1. Its prime factorisation contains only the two smallest odd prime numbers.

2. It has exactly $6$ distinct positive divisors.

3. It is strictly less than $100$ ($N < 100$).

1. a) Algebraically deduce the possible forms that the prime factorisation of any number having exactly $6$ divisors and only two distinct prime factors $p$ and $q$ can take. Recall that if a number factorises as $p^a \cdot q^b$, its total number of divisors is $(a + 1) \cdot (b + 1)$.
2. b) Identify which two specific prime factors $p$ and $q$ are in our problem.
3. c) Find all natural numbers $N$ that satisfy all the given conditions. For each solution you find, check the result by writing the ordered list of all its divisors.

The head of security at a community centre programs the three-digit numeric code $C$ for the sports equipment storage room. To make it easy to remember by mathematical deduction, the code has been designed with the following clues:

1. The code $C$ is the product of three consecutive prime numbers: $C = p_1 \cdot p_2 \cdot p_3$ (ordered from smallest to largest).

2. The sum of these three consecutive prime numbers is exactly $23$, that is, $p_1 + p_2 + p_3 = 23$.

3. If we add $5$ units to the code number, the new number obtained ($C + 5$) is simultaneously divisible by $2$, by $3$, and by $5$, but it is not divisible by $9$.

1. a) Set up an orderly search among the prime numbers to find $p_1$, $p_2$, and $p_3$. From them, calculate the numerical value of the code $C$.
2. b) Verify that the value obtained for $C$ strictly satisfies the third clue by analysing the number $C + 5$ using divisibility tests (without doing long divisions).
3. c) Reason, using the exponents of the prime factorisation of $C$, how many divisors the code has in total. Is it a prime or a composite number?