Multiples and Factors

A grasshopper is training for a competition. In each jump it advances exactly $6$ decimeters.

Write the total distance it has covered after taking $1, 2, 3, 4, 5$ and $10$ jumps. What are these numbers called with respect to $6$?

A bus route passes through the shopping center every $12$ minutes. If a bus has just left right now (minute $0$), write down all the minutes when the bus will pass again over the next two hours.

(Hint: You must find the multiples of $12$ until you reach $120$).

A well-known video game streamer uploads a clip to her social media exactly every $8$ hours. If today she uploaded the first clip at $08:00$ in the morning:

1. a) Write the sequence of times of the day (in 24-hour format) when she will publish her next $5$ clips.
2. b) Is the number of hours elapsed since the first clip a multiple of $8$? Justify your answer.
3. c) How many full days will have passed when she has published a total of $10$ clips (including today's)?

We have bought $18$ cans of soda for a party and we want to place them on the table forming complete rectangles (with no cans left over). For example, we can put $3$ rows of $6$ cans.

Find all the other possible ways to organize the cans and write the complete list of the factors of $18$.

Identify which of the following numbers are divisors of $24$ by carrying out the necessary divisions:

Remember: A number is a divisor of another if, when dividing them, the remainder is equal to $0$.

An animal shelter has received a donation of $48$ cans of cat food. They want to store them in small boxes, so that all the boxes have the same number of cans and none are left over.

1. a) Write all the factors of $48$.
2. b) If the manager wants each box to contain more than $5$ cans but fewer than $15$, what packaging options does he have? Indicate the number of cans per box and how many boxes he would need in each case.

Emma wants to share $28$ stickers equally among her $4$ friends.

1. a) How many stickers will each one receive? Are there any left over?
2. b) Is $28$ divisible by $4$?
3. c) Is $4$ a divisor of $28$?

Knowing that the equality $15 \cdot 4 = 60$ is true, complete the following sentences using the words multiple or factor correctly:

1. a) The number $60$ is a ____ of $15$.
2. b) The number $4$ is a ____ of $60$.
3. c) The number $60$ is a ____ of $4$.
4. d) The number $15$ is a ____ of $60$.

Look at the following "number jump" diagram and complete the sentences using the words multiple, factor or divisible:

1. a) $60$ is a .................... of $12$.
2. b) $12$ is a .................... of $180$.
3. c) $180$ is .................... by $60$.
4. d) If we add the number $5$ to the diagram, what relationship would it have with $60$?

In a comic shop, protective sleeves are sold in packs of $10$ units.

1. a) Can you buy exactly $30$ sleeves? And $45$? Explain why.
2. b) Write down the first seven multiples of the number $10$.

State whether the following statements are True or False and justify your answer with an example or a division:

1. a) The number $1$ is a divisor of all natural numbers.
2. b) Any number is a multiple of itself.
3. c) The number $8$ is a divisor of $4$.
4. d) A number can have infinitely many multiples.

Reason whether the following statements are true or false, giving an example or a counterexample in each case:

1. a) If a number is a factor of $10$, it is also a factor of all multiples of $10$ (such as $20, 30, 40...$).
2. b) The set of factors of a number is infinite.
3. c) If a number $a$ is a multiple of $b$, then $b$ is a factor of $a$.
4. d) Any natural number is, at least, divisible by itself and by one.

A graphic designer is organizing a gallery of $30$ identical square icons for a website. He wants to arrange them to form a complete rectangle (with no icons left over).

1. a) Find all the possible ways to organize the icons (for example: $1$ row of $30$).
2. b) What mathematical relationship do the dimensions of these rectangles have with the number $30$?
3. c) If the design only allows the shortest side of the rectangle to have at least $3$ icons, how many options does he have now?

In Art class we are creating patterns with square tiles. To make a "super square", we need the total number of tiles to be a perfect square.

1. a) If we have a tub with $80$ tiles, how many tiles are left over if we want to form the largest possible square?
2. b) How many tiles would we be short of, at least, to be able to form the next larger square?

Two LED lights on a stage flash at different rates. The blue light turns on every $6$ seconds and the orange light every $9$ seconds.

1. a) Write the first $10$ multiples of each number (starting with $0$).
2. b) At which seconds will both lights turn on at the same time during the first minute (from $0$ to $60$ seconds)?
3. c) What is the first second (after $0$) when they coincide? What do we call this number in mathematics?

In a Smart City, three smart lighting systems operate on different cycles:

• The park lights flash every $18$ seconds.

• The fountain lights change every $24$ seconds.

• The information panel updates every $30$ seconds.

If the three systems coincided for the first time at $20:00$, how many times will they coincide again before $20:15$? Use the relationship between the multiples of these numbers to work it out.

Two athletes are training on a circular track. The first takes $75$ seconds to complete one lap and the second takes $90$ seconds.

If they start running at the same time and in the same direction:

1. a) How many minutes will it take for them to coincide again at the starting line?
2. b) At that moment, how many laps will each have completed?
3. c) If the training session lasts a full hour, how many times will they coincide at the start, including the beginning and the end?

A famous influencer has received a batch of cosmetic products to raffle among her followers. She has $72$ lipsticks, $108$ nail polishes and $180$ face masks.

She wants to prepare the largest possible number of identical boxes (with the same contents in each) without any products left over.

• How many boxes can she prepare?

• What quantity of each product will be in each box?

• If she decides that each box cannot have more than $15$ items in total, how many boxes should she prepare now to continue ensuring they are all identical and nothing is left over?

An events company wants to organize a concert in a rectangular hall. They have bought a total of $150$ chairs. The organizer wants to arrange them so that all rows have the same number of chairs, but with the following restrictions:

• There must be more than $5$ rows.

• There must be more than $10$ chairs in each row.

• The number of chairs per row must be a factor of $450$ but cannot be a multiple of $4$.

Find all the possible ways to organize the chairs (rows and chairs per row) that meet these conditions.

Consider a natural number $n$ that satisfies the following two properties:

1. $n$ is a common multiple of $12$ and $15$.

2. $n$ is a factor of $360$.

Write all the possible values of $n$. For each value found, indicate the smallest factor (other than $1$) and the smallest multiple of $n$ that is greater than $1\,000$.

Find the value of the unknown digits $A$ and $B$ in the number $5A2B$ knowing that:

• The number is divisible by $2$ and by $3$ simultaneously.

• The number is a multiple of $11$.

• The digit $A$ is a factor of $16$ and the digit $B$ is a multiple of $4$ (other than zero).

Reason step by step how you rule out the options until you find the correct number.

A cybersecurity expert is trying to crack a four-digit PIN code hidden on a server. The system gives him the following clues about the number:

• It is a palindromic number (it reads the same from left to right as from right to left).

• It is a multiple of $9$ and also a multiple of $5$.

• The sum of its four digits is a factor of $36$.

• The number is greater than $5\,000$.

What is the PIN code the expert is looking for? Reason whether there is more than one possible solution.

A group of archaeologists has found an inscription with a secret numerical code. The head of the expedition gives you these clues to decipher it:

• It is a number between $125$ and $160$.

• It is a multiple of $15$.

• It is not an even number.

What is the secret code? Reason your answer by ruling out the options that do not meet the three conditions.

We are looking for a very special number that we will call "$X$". We know the following:

• $X$ has exactly $4$ factors.

• $X$ is a two-digit number.

• If we add all its factors, the result is $32$.

Find the value of $X$ explaining the process. Hint: Think about what type of numbers have exactly 4 factors (for example, the product of two different prime numbers).