Exercises: Prime factorisation

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 factorization) and identify the missing numbers in the empty gaps so that it is correct:

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

Match each number with its corresponding prime factorization:

• $24$ • $\quad$ • $2 \cdot 3^2$

• $18$ • $\quad$ • $3^2 \cdot 5$

• $45$ • $\quad$ • $2^3 \cdot 3$

• $50$ • $\quad$ • $2 \cdot 5^2$

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 one row.

Look at the drawing and find the prime factorization of the total number of dots that make up the rectangle:

Without doing the multiplication, look at the following factorization of a secret number $n$:

$n = 2^2 \cdot 3 \cdot 5$

Indicate whether the following statements are true or false, justifying your answer:

1. a) The number $n$ is a multiple of $5$.
2. b) The number $n$ is divisible by $7$.
3. c) The number $3$ is a factor of $n$.
4. d) The number $n$ is even.

Lucy wants to organize her $28$ video games into identical boxes. For the organization to be "mathematically perfect", she has decided that the number of video games in each box must be a prime factor of $28$.

In how many ways can she organize her boxes and how many games would go in each one? (Hint: start by finding the prime factorization of $28$).

To organize a video game tournament, we need to break down the number of participants into their "building blocks" (prime factors). Find the prime factorization using the vertical line method for the following numbers:

1. a) $126$
2. b) $180$
3. c) $420$

Fill in the empty gaps in this factor tree for the number $120$. Remember that each branch must end in a prime number for the factorization to be complete. Then write the final result using powers.

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

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

Without needing to calculate the value of $N$, reason whether the following statements are true or false:

1. a) $N$ is an even number.
2. b) $N$ is divisible by $9$.
3. c) $10$ is a factor of $N$.
4. d) $N$ is a multiple of $25$.

A pastry chef has made $150$ artisan chocolates. He wants to store them in boxes so that each box has the same number of chocolates and none are left over.

1. a) Find the prime factorization of $150$.
2. b) Using the factors from the previous section, write down all the possible ways to pack the chocolates if each box must have between $10$ and $30$ units.

Find a number that satisfies the following three conditions simultaneously:

1. Its prime factorization only contains the prime numbers $2$ and $3$.

2. It is greater than $30$ but less than $50$.

3. It is a multiple of $8$.

Sometimes, to factor numbers ending in zero, we can use mental "shortcuts". For example, we know that $10 = 2 \cdot 5$.

Factor the following numbers using this trick to save time on the vertical line:

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

It is known that the number $A$ and the number $B$ have the following prime factorizations:

$A = 2^2 \cdot 3 \cdot 7$

$B = 2 \cdot 3^2 \cdot 11$

1. a) What is the decimal value of each number?
2. b) What prime factors do they have in common?
3. c) Would the number $A \cdot B$ be divisible by $77$? Justify your answer by looking at their factors.

Find all the factors of the number $N$ knowing that its prime factorization 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.

A mystery number $n$ has the following prime factorization: $n = 2^a \cdot 3^b \cdot 5^c$.

We know that:

• If we divide $n$ by $10$, the result is still divisible by $5$.

• If we divide $n$ by $18$, the result is no longer divisible by $3$.

• The number $n$ is a multiple of $8$, but if we divide it by $2$, it stops being a multiple of $8$.

Calculate the values of the exponents $a$, $b$ and $c$ and find the value of the number $n$.

An incomplete prime factorization has been left on the board in the math laboratory. 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 $3$.

A logistics company wants to manufacture the largest possible cubic boxes to transport three types of products that come in rectangular containers. The dimensions of the products are:

• Product A: $360$ in

• Product B: $480$ in

• Product C: $540$ in

Use prime factorization to find the side length of the cubic box (which must be the Greatest Common Factor) and express this measurement as a product of its prime factors. How many boxes of that size will fit in total if we add those of the three products?

Three gamers are playing an online video game. Their characters have special abilities that recharge at different times:

• The first character launches an attack every $2^3 \cdot 3$ seconds.

• The second character uses a shield every $2^2 \cdot 3^2$ seconds.

• The third character heals every $2 \cdot 3 \cdot 5$ seconds.

If at the start of the game at 18:00:00 all three have used their ability at the same time, how many seconds must pass for them to coincide again? Express the waiting time first as a prime factorization and then in minutes and seconds.

Find the smallest number that has exactly $6$ factors, knowing that only the first two prime numbers appear in its prime factorization. Explain your reasoning by analyzing the exponents of its factorization.

The area of a rectangular garden is $252$ $m^2$. If we know that the measurements of its sides (in feet) are natural numbers and that the longest side is a multiple of $3$ and $7$, but not of $2$, use the prime factorization of the number $252$ to find all the 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 $B$, give a reasoned answer to:

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 factorization of the number that results from working out $M.C.D.(A, B)$?

A four-digit number has the form $3A2B$. Knowing that:

• It is divisible by $9$.

• It is divisible by $10$.

Calculate the value of the digits $A$ and $B$. Once the number is found, find its complete prime factorization and determine which is its largest prime factor.