Exercises: Prime numbers

At a card collectors' convention, the stands have been numbered from $10$ to $25$. The organizers want to highlight the stands that have a prime number with an orange sign. Write down the list of all the stand numbers that should have the orange sign.

Look at the following numbers and identify which ones are composite, justifying your answer (indicating at least three factors for each one):

$15, 17, 21, 23, 27$

The Sieve of Eratosthenes works like a filter that only lets prime numbers pass 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 composite numbers filter.

Complete this "factor tree" to find the prime factorization of the number $12$. Remember that only prime numbers can be left on the final branches.

A video game programmer wants to create a "final boss" whose strength level is the product of the first three different prime numbers. What strength level will the monster have?

Prime numbers are sometimes called "the building blocks of mathematics" because all other numbers are built with them.

1. a) If you multiply $3 \cdot 3 \cdot 5$, what composite number do you get?
2. b) Is it possible to get the number $20$ by multiplying only prime numbers? Prove it by writing its factorization.

In PE class, the teacher wants to organize the $13$ students into perfect rows and columns to do an exercise (for example, $2$ rows of $6$ people).

1. a) Why can't he do this with $13$ students in any way other than a single row of $13$?
2. b) What type of number is $13$ according to its factors?

Two numbers are said to be coprime (or mutually prime) if their only common factor is $1$. Check if the numbers $8$ and $9$ are coprime by following these steps:

1. Write down all the factors of $8$.

2. Write down all the factors of $9$.

3. Circle the ones they have in common. Are they coprime?

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

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

Carry out the prime factorization of the following numbers using the vertical bar method. Express the final result using powers where possible:

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

Look at the following prime factorization of a "mystery" number $N$:

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

Without needing to calculate the value of $N$, answer with justification:

1. a) Is $N$ divisible by $10$?
2. b) Is $N$ a multiple of $18$?
3. c) Write down three divisors of $N$ that are not prime numbers.

In a mathematics class, the teacher says that to know if a number smaller than $100$ is prime, it is enough to check that it is not divisible by $2, 3, 5$ and $7$.

Justify why it is not necessary 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.

A logistics warehouse wants to organize $120$ boxes of soft drinks and $144$ boxes of juice on identical pallets, so that each pallet has the same number of boxes of each type and that this number is the maximum possible.

1. a) How many boxes of each type will there be on each pallet?
2. b) How many pallets will be needed in total?

Three subway lines coincide at the central station at $06:00$ in the morning. Line A passes every $12$ minutes, line B every $15$ minutes, and line C every $20$ minutes.

At what time will the three lines coincide again for the first time? How many times will they have coincided in total before $10:00$ in the morning?

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

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

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

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

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

Hint: Remember that if a number is $p^x \cdot q^y$, its total number of divisors is $(x + 1) \cdot (y + 1)$.

At a young programmers' convention, they want to organize the attendees into identical teams. There are $154$ boys and $132$ girls. The goal is for each team to have the same number of boys and the same number of girls, and for the number of teams to be the maximum possible to encourage competitiveness.

1. a) How many teams will be formed and how many people will there be in each one?
2. b) If for safety reasons, the teams cannot have more than $20$ people, what is the next maximum number of teams that could be formed while meeting the conditions of equality?

To check if the number $601$ is prime, a student claims that he only needs to divide it by the prime numbers smaller than $25$.

1. a) Justify why the student has chosen the number $25$ as the limit.
2. b) Carry out the necessary divisions following his 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?

In the robotics laboratory, three drones have been programmed to emit a light signal at different intervals. Drone Alpha flashes every $45$ seconds, drone Beta every $60$ seconds, and drone Gamma every $72$ seconds. If all three drones are turned on simultaneously at $10:00$ in the morning:

1. a) At what exact time will the three flashes coincide again for the first time?
2. b) How many times will the three drones have coincided before $11:00$ in the morning (not counting the initial flash)?

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 divisors are $1, 2, 3$ and $1 + 2 + 3 = 6$.

1. a) Check if the number $28$ is a perfect number.
2. b) The next perfect number is greater than $400$ and less than $500$. Knowing that it is a multiple of $16$ and $31$, 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 different 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 divisors does the number $N$ have? Justify your answer without listing them.

Two numbers are said to be "twin primes" if both are prime and their difference is exactly $2$ (such as $3$ and $5$, or $11$ and $13$).

1. a) Find all the pairs of twin primes between $50$ and $100$.
2. b) Is it possible for there to be three consecutive odd numbers that are all prime (prime triplets)? If the answer is yes, give an example. If it is no, explain why by reasoning about divisibility by $3$.

We have a piece of string of $210$ in and another of $330$ in. We want to cut them into pieces of the same length, so that they are as long as possible and nothing is left over.

1. a) What length will each piece be?
2. b) Once cut, we want to store all the pieces in boxes. Each box can only contain pieces from a single string (they cannot be mixed). If each box must have the same number of pieces, what is the maximum number of pieces we can put in each box so that no pieces are left over and we use the fewest possible boxes?