Multiples, divisors, and divisibility rules

A grasshopper is training for a competition. With each jump it travels forward exactly $6$ decimeters.

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

Knowing that the equality $15 \times 4 = 60$ 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 $60$.

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

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

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 $60$?

State whether the following statements are true or false, giving an example or 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 1.

Identify which of the following numbers are factors of $24$ by performing the necessary divisions:

Remember: A number is a factor of another if, when divided, the remainder is equal to $0$.

A graphic designer is arranging a gallery of $30$ identical square icons for a website. They want to arrange them into a complete rectangle (without any icons left over).

1. a) Find all possible ways to arrange 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 shorter side of the rectangle to have at least $3$ icons, how many options do they have now?

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 down 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 do they have? State the number of cans per box and how many boxes would be needed in each case.

A librarian has to place books on shelves where exactly $3$ books fit into each space. To quickly find out if they can fill the shelves without any books left over, they use the divisibility test for $3$.

Calculate the sum of the digits of the following numbers and state which ones are divisible by $3$:

1. a) $123$
2. b) $401$
3. c) $516$
4. d) $802$

Complete the following table in your notebook by writing YES or NO in each cell, applying the divisibility tests you have learned:

Replace the letter $A$ with a digit so that the resulting number is a multiple of $9$. Remember that the sum of its digits must be a multiple of $9$.

1. a) $2A5$
2. b) $71A$
3. c) $A08$

Find the value of the missing digit, represented by the letter $A$, in each of these numbers so that the given condition is satisfied:

1. a) $24A$ is divisible by $4$ and $3$ at the same time.
2. b) $51A$ is divisible by $11$.
3. c) $A72$ is divisible by $9$ but not by $10$.

A baker has baked a tray of cookies. They know they have more than $70$ cookies but fewer than $85$. If they pack them in boxes of $6$, none are left over. If they pack them in boxes of $9$, none are left over either.

1. a) How many cookies have they baked exactly?
2. b) Justify your answer using the divisibility tests for $2$ and $3$ to explain why the number is divisible by $6$.

At a video game convention, the number of attendees is a value between $700$ and $750$. Knowing that this number is divisible by $3$, by $11$, and by $2$ at the same time, exactly how many people attended the convention? Explain how you ruled out the other possibilities.

A hacker has left a clue to decrypt a server password. The password is the smallest four-digit number, $2A3B$, that simultaneously meets these three conditions:

• It is divisible by $2$ but not by $4$.

• It is divisible by $9$.

• It is divisible by $5$.

What is the server password? Justify your answer by applying the rules step by step.

A number is divisible by $4$ if its last two digits are zeros or form a multiple of $4$.

Look at the following years and say which ones would be "special years" because they are divisible by $4$:

$1996, 2000, 2010, 2022, 2024, 2030$

Apply the divisibility test for $11$ to determine which of the following numbers are multiples of $11$. For those that are not, state how much you should add to them so that they are:

1. a) $1,232$
2. b) $5,049$
3. c) $8,175$

Find the value of the unknown digits $A$ and $B$ in the number $5A1B$ knowing that:

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

• The number is a multiple of $11$.

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

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

Find all possible values for the digits $X$ and $Y$ such that the five-digit number $4X12Y$ is divisible by $4$ and by $9$ at the same time. Note that there may be more than one winning combination.

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

1. a) If we have a jar with $80$ tiles, how many tiles are left over if we want to form the largest possible square?

1. b) How many more tiles would we need, at minimum, to be able to form the next larger square?

An event company wants to organise a concert in a rectangular room. They have bought a total of $150$ chairs. The organiser 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.

• In each row there must be more than $10$ chairs.

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

Find all possible ways to arrange 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 down all possible values of $n$. For each value found, state its smallest factor (other than $1$) and the smallest multiple of $n$ that is greater than $1\,000$.

A three-digit palindromic number is one that reads the same from left to right as from right to left (for example, $525$). Find a three-digit palindromic number that is divisible by $11$ and such that, when adding its digits, the result is a multiple of $3$. Is it the only possible solution?

Determine, without performing long division, whether the number $N = 10^5 + 8$ is divisible by $2, 3, 4, 9$ and $11$. Justify each answer by stating the corresponding divisibility rule or property. (Hint: write the number with all its digits first).