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:

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

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 $50$.

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.

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) $715$

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.

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) $203$

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 $\sqrt{35X1}$.

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

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

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.

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 $2^3 \times 3^2$.

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

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.

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.

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 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.

At a station on the Mexico City Metro, 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.

1. a) How many minutes must pass before they coincide again for the first time?
2. b) At what exact time will that happen?

A food bank has received a donation of $120$ jars of pulses, $150$ liters of milk and $72$ cans 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 $a \times b = 6750$. 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.

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 $m.c.d.(A, B)$.

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

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

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

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.

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.