Exercises: Multiples, factors and divisibility tests

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

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 $60$.

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 $0$.

Aoife 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 $28$?

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:

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 $4$:

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

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) $A08$

In a 1st Year Mathematics exam, Seán has to classify several numbers according to divisibility rules and writes the following three statements on his answer sheet:

1. "The number $253$ is divisible by $3$ because its last digit is $3$".

2. "The number $410$ is divisible by $4$ because it ends in $0$, and since $10$ is even, it can be divided twice by $2$".

3. "The number $837$ is divisible by $9$ because the sum of its digits is $8 + 3 + 7 = 18$, and $18$ is a multiple of $9$".

Your task:

1. a) State, with reasons, which of Seán's three statements are incorrect. Explain what conceptual error he made in each of them and what the correct rule is that he should have applied according to the notes.
2. b) Apply the appropriate rules to the numbers $253$ and $410$ and state whether or not they are divisible by $3$ and by $4$, respectively.

To open the digital padlock of her gym locker, Sara must enter a four-digit code of the form $5\text{.}2A4$, where $A$ is an unknown digit (from $0$ to $9$). Sara remembers two conditions that her code meets:

• The number is divisible by $3$.

• The number is not divisible by $9$.

1. a) Write the sum of the digits of the number in terms of $A$ and find all possible values that the digit $A$ can take so that the number is a multiple of $3$.
2. b) From the values you have obtained, rule out, with reasons, those that make the number divisible by $9$. Which valid four-digit numbers remain?
3. c) Check whether any of the resulting valid numbers is also divisible by $4$ by applying the rule for the last two digits, without performing the complete division.

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?

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 $60$?

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.

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?

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.

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 $6$.

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) $8,175$

The Physical Education department of a secondary school is organizing a hiking trip for the $198$ students in 1st Year. They want to distribute them into equal groups, with a minimum of $10$ and a maximum of $25$ students per group, so that no student is left without a group and all groups have exactly the same size.

1. a) Estimate: Before looking for divisors, round $198$ to $200$ and calculate mentally approximately how many groups would be formed if groups of $10$, $15$, or $20$ students were made. From that estimate and using divisibility rules, reason why it is impossible for the groups to have exactly $10$ or $20$ students.
2. b) Calculate: Using the divisibility rules (for $2$, $3$, $5$, $9$, and $11$) and the prime factorization of $198$ that follows from them, analyze which numbers between $10$ and $25$ are exact divisors of $198$. Deduce all possible ways to organize the activity (indicating how many groups are formed and how many students are in each group).
3. c) Check: Reflect on your results: why is it not possible to form groups of $15$ students even though $198$ is divisible by $3$?

A cooperative oil mill in Kilkenny packages a special batch of $2\text{.}772\text{ litres}$ of extra virgin olive oil. They wish to market the entire batch in identical jugs of integer capacity (no decimals), with no oil left over or missing. The cooperative has standard commercial containers of the following capacities: $2\text{ L}$, $3\text{ L}$, $4\text{ L}$, $5\text{ L}$, $9\text{ L}$, $10\text{ L}$, and $11\text{ L}$.

1. a) Apply the unit's divisibility rules in detail to determine which of these containers can be used to package exactly the $2\text{.}772\text{ litres}$ without doing paper-and-pencil divisions.
2. b) If the cooperative decides to use the largest possible jugs among the valid ones from the previous list to save containers, which one should it choose? How many jugs will it fill in total?
3. c) A distributor requests packaging part of the batch in $6\text{-litre}$ jugs. Without performing the division $2\text{.}772 : 6$, justify whether it is possible to do so based on the divisibility properties you have already verified in section (a).

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 $1\,000$.

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.

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.

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.

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.

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).

To check whether the number $43\text{.}956$ is divisible by $4$, by $9$, and by $11$, three 1st Year students propose different approaches:

• Aoife (Method 1 - Traditional division): Prefers to perform the three paper-and-pencil divisions ($43\text{.}956 : 4$, $43\text{.}956 : 9$, and $43\text{.}956 : 11$) and check if the remainder is $0$.

• Seán (Method 2 - Divisibility rules): Applies the unit's theoretical rules: looks at the last two digits for $4$, sums the digits for $9$, and subtracts the sum of odd-positioned and even-positioned digits for $11$.

• Niamh (Method 3 - Smart decomposition): Observes that $43\text{.}956 = 44\text{.}000 - 44$. Since $44\text{.}000$ and $44$ are obvious multiples of $4$ and $11$, she deduces that their difference must be as well.

Your task:

1. a) Apply Seán's method step by step to verify the divisibility of $43\text{.}956$ by $4$, $9$, and $11$.
2. b) Develop Niamh's method to analyze divisibility by $4$ and by $11$. Then, try to use her decomposition to decide whether it is divisible by $9$: are $44\text{.}000$ or $44$ multiples of $9$? What happens in this case?
3. c) Compare the three methods: which is the fastest for this specific number? If you had to analyze the number $43\text{.}951$, would Niamh's method still be advantageous compared to Seán's? Justify which one you would choose as a general rule.

An Irish banking institution generates four-digit security codes of the form $C = A\text{.}32B$ (where $A$ represents the thousands digit and $B$ the units digit). For the system to accept the code, three conditions must be met simultaneously:

1. The number is divisible by $4$ and by $5$ at the same time.

2. The number is divisible by $11$.

3. The number is divisible by $3$, but is not divisible by $9$.

1. a) In an orderly manner, applying the divisibility rules, deduce the required value that the digit $B$ must take. Then, show that there is only one possible value for the digit $A$. What is the final code $C$?
2. b) The bank's computer scientist states: "Any four-digit number ending in the same two digits as $C$ will be divisible by $4$, regardless of what digits we put in the thousands and hundreds places". Relying on the theoretical explanation from the notes about the number $100$ ($100 = 4 \times 25$), justify whether this statement is true or false.
3. c) Without doing long divisions, justify with reasons whether the obtained code $C$ is divisible by $33$ and whether it is divisible by $60$ using the properties of multiples and divisors.