Exercises: Prime numbers

In a strategy video game, you need to classify the 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 organize the shelves of a school library, the librarian uses the rules of divisibility. Without performing long division, indicate 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 previous numbers. Look at the following chart of the numbers from $1$ to $20$:

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

In Math class we are learning to break down numbers into their fundamental building blocks: prime factors. Perform the prime factorization (using the "vertical line") of the following numbers:

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

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

Use its prime factorization to find all its natural divisors.

Hint: Remember that $1$ and the number itself are always divisors.

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

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

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

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 $50$. What numbers could the code be? Check your answer using the rules of divisibility.

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

A messaging app uses a security filter that only allows messages whose identifier is a prime number to pass. Classify the following identifiers into prime numbers or composite numbers, justifying your decision in each case by finding a divisor other than one 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 that represents the factorization of a natural number. Your task is to complete it and finally write the prime factorization of this number using powers of prime factors.

At a Chicago subway station, three lines coincide at $07:00$ in the morning. Line $L1$ passes every $12$ minutes, line $L2$ every $15$ minutes and line $L5$ every $20$ minutes.

1. a) How many minutes must pass for them to coincide again for the first time?
2. b) At what exact time will this meeting take place?

A food bank has received a donation of $120$ jars of beans, $150$ gallons of milk and $72$ cans of preserves. They want to prepare the greatest possible number of identical boxes to distribute to families in need, with nothing left over from any product.

1. a) How many boxes can they prepare as a maximum?
2. b) What quantity of each product will each of these boxes 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 these numbers without needing to calculate them separately.

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

1. a) Find all the 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 value of the numbers $A$ and $B$ from their prime factorizations and, subsequently, find the $m.c.m.(A, B)$ and the $m.c.d.(A, B)$.

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

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

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

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 checking 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 what these two prime numbers are, check with reasoning that they are prime using the square root method, and calculate the value of the security key $N$.

In a rhythmic gymnastics exhibition, there is a group of $p$ gymnasts. The choreographer tries to organize the athletes into rows and columns to form a perfect rectangle, but observes 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 that has exactly 8 divisors.

If it is known that $40 < p < 60$, identify all possible values of $p$, show that $p$ is a prime number and find all the possible ways to organize 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 the pairs of prime numbers whose sum is exactly $120$. To show that the chosen numbers are really prime, select the pair whose components are furthest apart and apply the rule of divisibility for prime numbers less than $\sqrt{n}$ for each of them.

An architect is designing a circular plaza and wants to place streetlights on the edge. The number of streetlights must be a prime number $n$. For the lighting to be aesthetic, the architect has discovered that if he subtracts 1 from the number of streetlights $(n-1)$, 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 the prime factorization of the composite number $(n-1)$ in a factor tree.

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 possible "special" number and the smallest possible one. Then, find out how many divisors each of these two numbers has without needing to list them all, using only the exponents of their prime factorization.

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

1. Apply the rules of divisibility 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 generates a prime number. Bear in mind that you only need to test for divisibility by prime numbers up to the integer part of $\sqrt{35X1}$.

Two comets pass close to Earth periodically. 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 to $p$. It is known that the last time they coincided was $77$ years ago and that this was the first time they were seen since historical records began.

If $p$ and $q$ are two-digit numbers, identify which numbers they are and calculate how many years will have to pass for them to coincide again.

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

1. a) Classify them into prime and composite.
2. b) For the composite numbers, indicate which of them has the greatest amount of distinct divisors.
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.