Multiples and divisors

In a video game shop, they want to organise a shelf with $30$ games such that all rows have the same number of games and none are left over. Use the divisibility relation to complete the following sentences:

1. a) If they put $5$ games in each row, there will be $6$ rows. Therefore, $30$ is a ________ of $5$.
2. b) Since the division $30 : 10$ has no remainder, we say that $10$ is a ________ of $30$.
3. c) The number $30$ is ________ by $2, 3$ and $5$ at the same time.

Apply the divisibility tests to state whether the number $4.350$ is divisible by the following values (answer YES/NO and briefly justify):

1. a) Is it divisible by $2$?
2. b) Is it divisible by $3$?
3. c) Is it divisible by $5$?
4. d) Is it divisible by $10$?

A tech shop has received a shipment of tablets. The manager knows that the total number of tablets is a four-digit number of the form $4A7B$. To organise the inventory, the number needs to satisfy the following conditions:

• It must be divisible by $5$ and by $2$ at the same time.

• It must be a multiple of $9$.

Determine all possible values of $A$ and $B$ so that the manager can organise the inventory.

Classify the following numbers from the list as prime or composite. Remember that composite numbers can be "broken down" as the product of other smaller numbers:

Explain why you chose the number $21$ as composite, stating which numbers it can be divided by besides $1$ and itself.

Complete the following "factor tree" to find the prime factorisation of the number $60$.

Once completed, write the final factorisation using powers: $60 = 2^{\dots} \cdot \dots \cdot \dots$

Two alarm clocks go off at different intervals. The first goes off every $15$ minutes and the second every $20$ minutes. If they have just gone off together right now at $08:00$ in the morning:

1. a) In how many minutes will they sound together again for the first time? (Calculate the $m.c.m.$).
2. b) At what exact time will they coincide again?

Marta has $24$ red stickers and $36$ blue stickers. She wants to make identical surprise envelopes for her friends, such that each envelope has the same number of stickers of each colour and none are left over.

1. a) What is the maximum number of envelopes she can prepare? (Calculate the $m.c.d.$).
2. b) How many stickers of each colour will be in each of those envelopes?

An esports club wants to organise a charity tournament. They have $120$ blue t-shirts, $180$ red t-shirts and $210$ white t-shirts. They want to prepare gift backpacks containing the same number of t-shirts of each colour, distributing all the t-shirts they have and without mixing colours within each individual package, while maximising the number of t-shirts per backpack.

1. a) What is the maximum number of t-shirts that can be in each backpack?
2. b) How many backpacks of each colour will be obtained?