Prime Factorization

Classify the following natural numbers into prime or composite. Remember that prime numbers are those that only have two divisors: $1$ and themselves.

$14, 17, 21, 23, 27, 29, 33, 37$

They want to design a special barcode based on a number $K$. This number is formed by multiplying the first $n$ prime numbers consecutively (for example, if $n=3$, the number would be $2 \cdot 3 \cdot 5 = 30$).

If we want the number $K$ to:

1. Be greater than $1000$ but less than $5000$.

2. Have exactly $6$ distinct prime factors in its prime factorization.

What is the value of $K$? Show the list of prime numbers you have used and check that the result is within the requested range.

State which number the following prime factorizations correspond to:

1. a) $2^{2} \cdot 3 \cdot 5$

1. b) $2^{3} \cdot 7$

1. c) $3^{2} \cdot 11$

What is the largest two-digit number that has only $2$ and $3$ as prime factors in its factorization?

Help example: the number $6 = 2 \cdot 3$ meets the condition, but it is not the largest two-digit number.

Complete the following factor tree for the number $24$. You must find two numbers that multiplied give the level above, until all the numbers at the ends are prime.

Finally, write the factorization of $24$ as a product of its prime factors.

Copy this prime factorization diagram into your notebook and fill in the empty gaps (marked with the letter $x$) with the corresponding numbers:

Fill in the gaps of the following prime factorization of a number using the vertical line method:

Write the final result of the factorization using powers.

Find the prime factorization of the following numbers. Express the final result as a product of powers:

1. a) $36$

1. b) $45$

1. c) $80$

Look at the prime factorization of the following numbers:

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

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

Without performing the multiplications, answer with reasoning:

1. a) Is the number $A$ a multiple of $7$?
2. b) Is the number $B$ divisible by $10$? (Hint: observe if it has the necessary factors to form $10$).
3. c) Is the number $3$ a divisor of $A$?

From the prime factorization of the number $150$, answer the following questions reasoning your answer:

1. a) Is $150$ divisible by $10$?
2. b) Is $150$ divisible by $4$?
3. c) Is $150$ divisible by $15$?

Find the number whose prime factorization is $2^3 \cdot 3^2 \cdot 11$. Once found, explain whether that number is a multiple of $6$ and $22$ by looking only at its prime factors (without performing divisions).

A Physical Education teacher wants to organize their $28$ students into equal groups of more than $2$ people but fewer than $10$, so that no student is left over.

Use the prime factorization of $28$ to find all the possible ways of making the groups.

A professional Instagram photographer wants to organize their photo feed. They have $210$ images and want to post them in equal blocks.

1. a) Find the prime factorization of $210$.
2. b) Write down four different ways of grouping the photos (for example: "3 blocks of 70 photos") using combinations of their prime factors.

A logistics company uses small cubic boxes to store parts. The total volume of a cargo container is $151200$ $cm^{3}$. The container must be completely filled with rectangular packages measuring $a$ in wide, $b$ in long and $c$ in high, where $a$, $b$ and $c$ are distinct prime numbers.

If we know that the container can hold exactly a number of packages that is a multiple of $10$:

1. Find the prime factorization of $151200$.

2. Identify the possible dimensions of the packages if the volume of each package must be greater than $100$ $cm^{3}$.

A video game designer wants to organize a virtual warehouse of items. To do this, they need to know how many divisors the number of available boxes has, which is $360$.

1. a) Find the prime factorization of $360$.
2. b) Using the rule of exponents (add one to each exponent of the factorization and multiply them), calculate the total number of divisors that $360$ has.

A 7th grade student claims to have found a number $X$ whose prime factorization is $2^{2} \cdot 3^{4} \cdot 6^{2} \cdot 5$.

1. Explain why this expression is not a valid prime factorization.

2. Correct the expression so that it is a real prime factorization.

3. Without calculating the numerical value of $X$, determine how many divisors it has. Show the calculation process step by step.

A hacker has blocked access to a server and left the following clue about the numerical password, which is a four-digit number $N$:

• The prime factorization of $N$ is of the form $2^{a} \cdot 3^{b} \cdot 5^{c}$.

• The total number of divisors of $N$ is exactly $30$.

• The number is a multiple of $100$ but is not a multiple of $8$.

Determine with reasoning the value of the password $N$ and show its complete prime factorization.

Find the prime factorization of the numbers $720$ and $1260$ using the vertical line method. Once the expressions in power form are obtained, state which prime factors they have in common and what is the highest exponent with which each one appears in both factorizations.

Consider two natural numbers, $A$ and $B$, whose prime factorizations are as follows:

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

$B = 2^{y} \cdot 3^{2} \cdot 7^{1}$

Knowing that the greatest common factor of both is $M.C.D.(A, B) = 36$, find the value of the unknowns $x$ and $y$. Once found, calculate the value of the least common multiple $m.c.m.(A, B)$ also expressed in its prime factorization form.

In a satellite network, Satellite A completes an orbit every $1260$ minutes, Satellite B every $1584$ minutes and Satellite C every $n$ minutes.

It is known that the last time the three coincided in the sky over Chicago was exactly the time indicated by their $m.c.m$ ago.

If the prime factorization of $n$ is $2^{3} \cdot 3^{2} \cdot 7$, calculate:

1. The prime factorization of the times of satellites A and B.

2. The total time (in minutes) that must pass for the three satellites to coincide again. Leave the result expressed as a product of powers of prime factors.

We have three natural numbers decomposed into prime factors:

$n_1 = 2^{3} \cdot p \cdot 7$

$n_2 = 2^{q} \cdot 3^{2} \cdot 5$

$n_3 = 3 \cdot 5^{2} \cdot 11$

If we know that:

• $M.C.D.(n_1, n_2) = 8$

• $m.c.m.(n_2, n_3) = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11$

Find the values of the unknown prime numbers or exponents $p$ and $q$. Justify your answer using the properties of prime factorization for calculating the $M.C.D.$ and the $m.c.m$.

We say that a number is a perfect square if all the exponents of its prime factorization are even numbers.

1. a) Find the prime factorization of the number $180$. Is it a perfect square?
2. b) What is the smallest number by which you would have to multiply $180$ so that the result is indeed a perfect square? Justify your answer by analyzing the exponents.

The volume of a cube-shaped box is exactly $1728$ in$^3$.

1. a) Find the prime factorization of the number $1728$.
2. b) Knowing that the volume of a cube is calculated as $V = l^3$ (where $l$ is the edge), use the previous factorization to group the factors into three identical groups and find the measurement of the edge.

Find out which is the smallest number that simultaneously meets these three conditions:

• It is a perfect square (all the exponents of its prime factorization are even).

• It is a multiple of $42$.

• It is a multiple of $60$.

To solve it, use the prime factorization of $42$ and $60$ and reason which factors and exponents you must add to obtain the minimum perfect square.