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1. Create a root node that contains the number $54.$
2. From the root node, draw two branches that a factor pair of $54.$
3. Break each factor of $54$ into its own factor pair.
4. Continue this process until only there are only prime factors on each branch.

With this information in mind, the factor tree of $54$ can now be created. Now, consider that the missing value on the red cell is the most repeated prime factor of the prime factorization of $54.$ Here, the number $3$ is repeated three times, while the number $2$ appears only once. This means that the missing value on the red cell is $3.$ In this case, the prime factorization of $64$ contains only the number $2$ six times. Therefore, the missing value on the yellow cell of the Sudoku game is $2.$ Magdalena can now fill the yellow and red cells in the Sudoku board!

The clue must be solved to help Magdalena figure out the number that goes in the purple cells. The clue claims that the number in the cells is given by the greatest common factor of $35$ and $85.$ To find the GCF, these steps can be followed.

1. Find the prime factorization of $35.$
2. Find the prime factorization of $84.$
3. Identify the common prime factors between the factorizations of $35$ and $84.$
4. Multiply the common prime factors.

Each of these steps will now be applied.

Prime Factorization of $35$

The prime factorization of $35$ can be found by using a factor tree. The root node of the tree is $35$ and then factor pairs of $35$ will extend from the root node. The prime factorization of $35$ is $5\cdot 7.$

Prime Factorization of $84$

A similar process can be followed to find the prime factorization of $84.$ The prime factorization of $84$ is $2\cdot 2\cdot 3\cdot 7.$

Identify Common Prime Factors

Now that the two prime factorizations are stated, find the common prime factor shared by these factorizations. The only common prime factors shared by these factorization is the number $7.$

Multiply

In this case, only one factor is shared by the prime factorizations of $35$ and $84,$ so the greatest common factor of $35$ and $84$ is $7.$ This number goes into the purple cells.

What a great achievement! Magdalena has made great progress on her puzzle.

Before finding the numbers to fill in the blue and pink cells, Magdalena first needs to find the least common multiple (LCM) of $10$ and $16.$ Consider these steps to find the LCM.

1. Determine the prime factorization of $10.$
2. Determine the prime factorization of $16.$
3. Write the prime factorization in a table to match prime factors vertically.
4. Bring down the primes in each column and multiply them to get the LCM.

Next, each of these steps will be performed.

Prime Factorization of $10$ and $16$

Both prime factorizations will be found using factor trees. One tree will have $10$ as its root, while the root of the second tree will be $16.$ Two branches extend from each root node to connect a factor pair. The process is repeated with the factors until only prime factors are on each branch.

Math Prime Factors Vertically

Now that the prime factorizations of $10$ and $16$ are found write them in a table. In the first row, place the prime factorization of $10$ and the factorization of $16$ in the second row. The table's columns will contain each factor while matching them vertically when possible.

Bring Down the Primes in Each Column and Multiply

Lastly, bring down the factors of $10$ and $16$ in each column of the table created previously.

The product of these prime factors is the least common multiple of $10$ and $16.$

Factors	Multiplication	$\text{LCM}(10,16)$
$2,$ $2,$ $2,$ $2,$ and $5$	$2\cdot 2\cdot 2\cdot 2\cdot 5$	$80$

The least common multiple of $10$ and $16$ is $80.$

Finding the Numbers to Fill in the Orange and Pink Cells

Magdalena found that the $\text{LCM}(10,16)$ is $80$ but she is not ready to fill in the cells in the puzzle yet. For the orange cells, she needs to find the number that gives $80$ when multiplied by $10.$ The number that goes in the pink cells can be found in a similar way. This means that the number that goes in the orange cells is $8$ and the number that goes in the pink cells is $5.$ What a big step Magdalena has made now in her solution process!

First, find the prime factorizations of these numbers using a factor tree. Now that the prime factorizations have been found, look for the common prime factors between them. The only common factor between the factorizations is $7,$ so the $\text{GCF}(49,84)$ is $7.$ This means that each bouquet will contain exactly $7$ flowers! Let $c$ be the greatest possible number of flowers per bouquet. Next, let $a$ be the number of rose bouquets and $b$ the number of tulip bouquets. Note that the sum of $a$ and $b$ represents the total number of bouquets. Additionally, multiplying the total number of bouquets by the number of flowers per bouquet $c$ will equal the number of flowers in total. It was previously determined that there are $7$ flowers per bouquet. The Distributive Property can be applied to this expression. Now, consider that the number of rose bouquets $a$ times seven will equal $49,$ the total number of roses. The missing value of $a$ can be found using this information. The number of tulip bouquets $b$ can be determined by following a similar procedure. This means that Magdalena and her mother can create $7$ rose bouquets and $12$ tulip bouquets.

1. Each group must contain the same number of juniors.
2. Each group must contain the same number of sophomores.
3. Each group must contain the same number of seniors.

There are $100$ juniors, $88$ sophomores, and $76$ seniors in the club. The coach wants to divide each grade level evenly to create the greatest number of groups possible. In other words, he wants to find the greatest common factor of these numbers. First, list the factors of each number.

Number	Factors
$100$	$1,$ $2,$ $4,$ $5,$ $10,$ $20,$ $25,$ $50,$ $100$
$88$	$1,$ $2,$ $4,$ $8,$ $11,$ $22,$ $44,$ $88$
$76$	$1,$ $2,$ $4,$ $19,$ $38,$ $76$

Notice that these numbers have the common factors $1,$ $2,$ and $4.$ Their greatest common factor is then $4.$ This means that the greatest number of groups that the coach can make is $4.$ This is also the number for the blue cells of Magdalena's puzzle — $4.$

The same thought can be applied to Vincenzo's and Tadeo's situations. This means that the least common multiple of $6,$ $10,$ and $5$ will give the number of days until the boys visit the cinema on the same day again. List the multiples of each to find the $\text{LCM}(6,10,5).$

Number	Multiples
$6$	$6,$ $12,$ $18,$ $24,$ $30,$ $36,$ $42,$ $\dots$
$10$	$10,$ $20,$ $30,$ $40,$ $50,$ $60,$ $70,$ $\dots$
$5$	$5,$ $10,$ $15,$ $20,$ $25,$ $30,$ $35,$ $\dots$

The least common multiple of $6,$ $10,$ and $5$ is $30.$ This means it will take $30$ days for the boys to visit the cinema on the same day. Now, recall that Tadeo goes to the cinema every five days. This means that $5$ times a certain number must equal $30.$ This number is $6.$ This means that Tadeo will go to the cinema $6$ times before meeting his friends. This is the solution to the riddle, so Magdalena can fill in the green cells with the number $6.$

Great news — the remaining numbers are easy to decipher and Magdalena has now filled in the rest of the board. What an outstanding achievement!