Exercises: Prime numbers

In a strategy video game, you need to classify energy crystals according to their purity. Crystals with prime numbers give extra energy, while composite numbers must be processed.

Classify the following numbers in the table in your notebook:

$11, 15, 23, 27, 31, 39, 42, 47, 51$

To organise the shelves of a school library, the librarian uses divisibility tests. Without carrying out long division, state which of the following numbers are divisible by $3$ and which by $5$:

1. a) $123$
2. b) $405$
3. c) $1.020$
4. d) $715$

The Sieve of Eratosthenes is an ancient method for finding prime numbers by crossing out the multiples of earlier numbers. Look at the following grid of numbers from $1$ to $20$:

Copy the grid into your notebook and complete the process up to the number $50$ to identify all prime numbers less than $50$.

In Maths class we are learning to break numbers down into their basic building blocks: prime factors. Find the prime factorisation (using a vertical bar) of the following numbers:

1. a) $24$
2. b) $60$
3. c) $100$

A programmer is creating a list of all the factors of the number $30$ for a security algorithm.

Use its prime factorisation to find all of its natural factors.

A candy shop wants to pack $20$ strawberry lollipops and $28$ lemon lollipops into identical bags, so that they have the maximum possible number of lollipops without mixing them. To do this, they must calculate the Greatest Common Divisor (G.C.D.).

In the figure, each division of the bars represents one lollipop.

Calculate the G.C.D. of $20$ and $28$ and explain how many lollipops there will be in each bag.

Two friends are running on a circular track. Ana takes $8$ minutes to complete one lap and Borja takes $12$ minutes. If they start together from the finish line, how many minutes will pass before they meet again at the finish line for the first time?

Use the lowest common multiple (LCM) to solve it.

Did you know that prime numbers are used to create security codes on the internet? A secret code is a prime number greater than $40$ and less than $50$. Which numbers could be the code? Check your answer using divisibility tests.

Marcos states in Maths class: «The number $91$ is prime because I checked that it is not divisible by $2$ (it is odd), nor by $3$ (its digits add up to $10$), nor by $5$ (it does not end in $0$ or $5$). Since I have already tested the first primes, there is no need to continue».\na) Explain the criterion that indicates up to which prime number it is necessary to test divisions to determine with certainty whether a number $N$ is prime.\nb) Up to which prime number should Marcos have tested for the number $91$?\nc) Find Marcos's mistake, explain why his deduction is false, and determine whether $91$ is prime or composite, justifying your answer with a division.

Before doing any calculations on paper, focus on the number $72$:\na) Estimate: Do you think $72$ will have more or fewer than $8$ divisors? Write down at least four divisors that immediately come to your mind to support your initial estimate.\nb) Calculate: Find the prime factorisation of $72$ and apply the formula for the number of divisors, $(a+1)(b+1)\dots$, to find the exact quantity.\nc) Check: Compare the obtained value with your previous estimate and write a brief conclusion explaining whether your initial intuition was correct and why a number less than $100$ can have that number of divisors.

In an online strategy game, to unlock a chest you need to enter a four-digit passcode of the form $52X4$. If the system tells you that the passcode is divisible by $3$ and $4$ simultaneously, what are all the possible values that the digit $X$ can take? Give reasons for your answer using the corresponding divisibility tests.

A messaging app uses a security filter that only allows messages through if their identifier is a prime number. Classify the following identifiers as prime numbers or composite numbers, justifying your decision in each case by finding a factor other than 1 and the number itself:

1. a) $119$
2. b) $131$
3. c) $143$
4. d) $151$
5. e) $203$

Look at the following incomplete factor tree representing the prime factorisation of a natural number. Your task is to complete it and finally write down the factorisation of that number using powers of prime factors.

At a metro station in Madrid, three lines coincide at $07:00$ in the morning. Line $L1$ runs every $12$ minutes, line $L2$ every $15$ minutes and line $L5$ every $20$ minutes. The figure shows the departures of each line during the first $40$ minutes from $07:00$.

1. a) How many minutes must pass for them to coincide again for the first time?

1. b) At what exact time will this meeting take place?

A food bank has received a donation of $120$ jars of pulses, $150$ litres of milk and $72$ tins of food. They want to prepare the maximum possible number of identical boxes to distribute to families in need, without any of the products left over.

1. a) What is the maximum number of boxes they can prepare?
2. b) How much of each product will each box contain?

It is known that the highest common factor of two numbers is $m.c.d.(a, b) = 15$ and that their product is $a \cdot b = 6750$. Use the mathematical relationship between the product of two numbers and their common factors/multiples to find the lowest common multiple of those numbers without calculating them separately.

Pairs of prime numbers that are separated only by an even number (that is, their difference is $2$) are called twin prime numbers. For example, $3$ and $5$ are twin primes.

1. a) Find all pairs of twin prime numbers less than $60$.
2. b) Is the pair $(21, 23)$ a pair of twin primes? Justify your answer.

Calculate the numerical values of the numbers $A$ and $B$ from their prime factorisations and then find the $m.c.m.(A, B)$ and the $m.c.d.(A, B)$.

$A = 2^{3} \cdot 3^{2} \cdot 5$

$B = 2^{2} \cdot 3 \cdot 5^{2}$

State whether $A$ and $B$ are coprime and explain why based on the result of the highest common factor.

To check whether the number $173$ is prime, two classmates propose two different strategies:\n- Lucía's strategy: Divide $173$ by all consecutive odd numbers starting from $3$ up to half of the number ($86$), that is: $3, 5, 7, 9, 11, 13, 15, \dots, 85$.\n- Guille's strategy: Mentally calculate which integers $\sqrt{173}$ lies between, and test dividing $173$ only by the prime numbers that are less than or equal to that square root.\na) Compare both strategies and justify why Guille's saves effort without losing reliability. In particular, explain why there is no need to test composite numbers such as $9$ or $15$, and why it is safe to stop at the square root.\nb) Apply Guille's strategy by carrying out the necessary divisions and conclude whether $173$ is prime or composite.

In a municipality in the province of Cuenca, the electoral roll for an auxiliary polling station registers exactly $105$ voters called to vote in local elections. The coordinator needs to organise the voters into groups or time slots that have exactly the same number of people, with the conditions that each slot must have more than one voter and that there cannot be a single slot with all the voters.\na) Decompose $105$ into prime factors and use the formula $(a+1)(b+1)(c+1)\dots$ to calculate how many divisors it has in total.\nb) Deduce how many different ways there are to organise the slots under the coordinator's conditions and write all the possible pairs of (number of slots, voters per slot).\nc) If $4$ more people register at the last minute and the electoral roll becomes $109$ voters, would it be possible to organise equal slots under the same conditions? Apply the method of primes up to $\sqrt{N}$ to check whether $109$ is prime and justify the organisational consequences.

A cybersecurity expert is designing an encryption system for a messaging app. The main security key is a number $N$ obtained by multiplying a pair of twin prime numbers between $100$ and $110$. However, upon reviewing the list, he realizes that there are two pairs of twin primes in that interval, so the choice is not unique. To solve the problem, he decides that the key will be the product of the two prime numbers closest to $105$ (one below and one above).

Determine which two prime numbers these are, prove with reasons that they are prime using the square root method, and calculate the value of the security key $N$.

In a rhythmic gymnastics display, there is a group of $p$ gymnasts. The choreographer tries to organise the athletes in rows and columns to form a perfect rectangle, but notices that the only way to do so is by placing them all in a single row. By adding one more gymnast to the group, the total number of people $(p+1)$ becomes a number with exactly 8 factors.

Given that $40 < p < 60$, identify all possible values of $p$, prove that $p$ is a prime number, and find all possible ways to organise the group of $(p+1)$ gymnasts into rectangles of more than one row.

Goldbach's conjecture states that every even number greater than 2 can be written as the sum of two prime numbers. Your task is to investigate this conjecture for the number $120$.

Find and write down all pairs of prime numbers whose sum is exactly $120$. To prove that the chosen numbers are indeed prime, select the pair whose components are furthest apart from each other and test for divisibility by prime numbers less than $\sqrt{n}$ for each of them.

An architect is designing a circular square and wants to place street lamps along its edge. The number of street lamps must be a prime number $n$. To make the lighting aesthetically pleasing, the architect has discovered that if you subtract 1 from the number of street lamps $(n-1)$, the result must be a number whose prime factorization is $2^3 \cdot 3^2$.

Calculate the value of $n$, justify why it is a prime number, and represent in a tree diagram the prime factorization of the composite number $(n-1)$.

You can do it on paper and send a photo with your answer.

A natural number is said to be "special" if it is the product of three distinct prime numbers less than $20$. An example would be $2 \cdot 3 \cdot 5 = 30$.

Find the largest and smallest possible "special" numbers. Next, find how many factors each of these two numbers has without needing to list them all, using only the exponents of their prime factorisations.

Find the values of the digit $X$ such that the number $35X1$ is a prime number. To do this, follow these steps:

1. Apply the divisibility rules to rule out values of $X$ that make it divisible by $3$ or $11$.

2. From the remaining values, use the square root method to check which of them generate a prime number. Keep in mind that you only need to test divisibility by prime numbers up to the integer part of $\sqrt{35X1}$.

Two comets periodically pass close to Earth. Comet A passes every $p$ years, where $p$ is a prime number. Comet B passes every $q$ years, where $q$ is the next consecutive prime number after $p$. Comet A passes today, and it is known that the last time they coincided was $77$ years ago, which was the first time they were seen together in recorded history.

If $p$ and $q$ are two-digit numbers, identify which numbers they are and calculate how many years must pass before they coincide again.

Consider the natural numbers between $90$ and $100$.

1. a) Classify them as prime or composite.
2. b) For the composite numbers, state which of them has the greatest number of distinct factors.
3. c) Is there any number in that interval that is the result of raising a prime number to a power greater than 1? Justify your answer.

A natural number $N$ has exactly $8$ divisors, and in its prime factorisation only $2$ and $3$ appear as prime factors.\na) Knowing that the formula for the number of divisors is $(a+1)(b+1) = 8$, state which pairs of non-zero natural exponents $(a, b)$ can $2$ and $3$ have so that the result is exactly $8$.\nb) From the pairs found, write down the possible prime factorisations and calculate the numerical values that the number $N$ can take.\nc) Which of these numbers is the smallest? Explain with reasoning how placing the larger exponent on the smaller base ($2$) or on the larger base ($3$) affects obtaining the smallest possible value.

Analyse the relationship between the exponents in the prime factorisation of a natural number and the total number of its divisors:\na) A natural number is a «perfect square» if it comes from squaring another natural number (for example, $36 = 6^2$). Prove with reasoning, using the formula $(a+1)(b+1)\dots$, why every perfect square necessarily has an odd number of divisors, whereas numbers that are not perfect squares have an even number of divisors.\nb) Following an inverse process, determine what is the smallest natural number that has exactly $9$ divisors. Show step by step how you deduce which prime factors and which exponents it must have to be the minimum possible.\nc) Check whether the number you obtained in part b) is a perfect square or not, and justify why this fact is consistent with your proof in part a).