Exercises: Divisibility

On a streaming platform, series are grouped into seasons. If a series has a total of $42$ episodes, state which of the following statements are true and which are false, giving a brief reason for your answer:

1. a) The number $6$ is a factor of $42$.
2. b) $42$ is a multiple of $5$.
3. c) $42$ is divisible by $14$.

Calculate mentally and write down in your notebook:

1. a) The first five multiples of $12$.
2. b) All the factors of the number $24$.
3. c) The multiples of $7$ between $30$ and $60$.

Look at the numbers appearing on the following board and classify them into prime numbers or composite numbers.

A video game requires you to enter a four-digit code to unlock a level: $3.45\_$. Help the player find the last digit so that the number satisfies the following conditions:

1. a) The number is divisible by $2$ and $5$ at the same time.
2. b) The number is divisible by $3$, but not by $2$.
3. c) It is a multiple of $9$.

Find the prime factorisation of the following natural numbers. Remember to write the final result as a product of powers:

1. a) $36$
2. b) $60$
3. c) $126$

Calculate the Highest Common Factor (HCF) and the Lowest Common Multiple (LCM) of the following pairs of numbers:

1. a) $10$ and $15$
2. b) $20$ and $30$
3. c) $8$ and $12$

Two lights on a gaming keyboard flash in different ways. The blue light turns on every $6$ seconds and the orange light every $9$ seconds. If both turned on at the same time right now, how many seconds will pass before they flash together again for the first time?

Aoife has $24$ star stickers and $16$ planet stickers. She wants to make identical gift envelopes (with the same number of stars and the same number of planets in each) without leaving any stickers over.

What is the maximum number of envelopes she can prepare? How many stickers of each type will there be in each envelope?

Cian has to check on an exam whether the number $3,522$ is divisible by $6$. In his work, he writes the following: “Since the last digit is $2$, the number is divisible by $2$. However, since that last digit ($2$) is not divisible by $3$, the number $3,522$ cannot be divisible by $6$”.

1. a) Identify the conceptual error Cian makes when checking for divisibility by $3$ and explain why his reasoning is incorrect.
2. b) Correctly apply the divisibility rules for $2$ and $3$ to determine whether or not $3,522$ is divisible by $6$.

A municipal printing house in Kilkenny has produced $1,342$ calendars, and the workers must pack them into boxes of exactly $4$ units each, with none left over.

1. a) Without performing long division or applying the full rule yet, make a quick estimate: knowing that $1,300$ and $40$ are multiples of $4$, explain whether you think it is reasonable that some calendars will be left over.
2. b) Apply the divisibility rule for $4$ by looking exclusively at the last two digits to check your estimate.
3. c) In light of the result, state how many calendars are left over after filling the maximum number of full boxes of $4$, or how many calendars would be needed to complete one more box.

A programmer is designing a four-digit access code. The code follows the format $35A2$, where $A$ is an unknown digit. For the code to be valid, the number must be simultaneously divisible by $3$ and $4$.

Determine all possible values that the digit $A$ can take and write down the resulting complete numbers.

At a video game convention, they want to organise the participants into teams. We have a security access code that is a four-digit number of the form $4A2B$.

For the code to be valid, it must satisfy two conditions:

1. a) It must be divisible by $3$.
2. b) It must be divisible by $11$.

Find all possible values for the digits $A$ and $B$.

Calculate the total number of factors of the number $360$ without writing them all out. To do this:

1. a) Find its prime factorisation using powers.
2. b) Apply the rule of multiplying each exponent increased by one.
3. c) State whether the number $360$ is a perfect square, explaining your reasoning based on the exponents in its factorisation.

An interior design company has two rolls of coloured LED strips: one blue light strip measuring $120$ metres and one warm light strip measuring $156$ metres. They want to cut both rolls into pieces of equal length, as long as possible, leaving no material left over.

1. a) How long should each piece be?
2. b) How many pieces of each colour will they get in total?

Three streaming channels start broadcasting special content today at 10:00 am.

• Channel A broadcasts a giveaway every $18$ hours.

• Channel B broadcasts an interview every $24$ hours.

• Channel C broadcasts a tournament every $30$ hours.

What is the minimum number of days that must pass before the three events coincide at the same time again? At what time and on which day of the week will this happen if today is Monday?

Find a natural number that simultaneously satisfies the following conditions:

1. It is between $200$ and $250$.

2. It is a multiple of $13$.

3. When divided by $2$ and by $3$, it gives an exact division.

Look at the following polygonal numbers representing pentagonal numbers. The first terms are $1, 5, 12, 22...$

1. a) Draw the pattern of dots for the fifth pentagonal number in your notebook.
2. b) Following the sequence, what is the numerical value of the fifth and sixth pentagonal numbers?
3. c) Check whether the tenth pentagonal number ($145$) is divisible by $3$, by $5$, and by $11$.

You can do it on paper and send a photo with your answer.

Given that the product of two numbers is $2160$ and their highest common factor (HCF) is $12$:

1. a) Calculate the lowest common multiple (LCM) of these numbers using the relationship between HCF and LCM.
2. b) If one of the numbers is $60$, what is the other number?
3. c) Find the prime factorisation of both numbers to check that the results from the previous parts are correct.

A natural number $N$ simultaneously satisfies the following three properties regarding its divisors and factors:

1. In its prime factorization, only $2$ and $3$ appear (both with an exponent greater than zero).

2. The total number of divisors of $N$ is exactly $9$.

3. $N$ is a perfect square.

1. a) Use the formula for the number of divisors and the property of the exponents of perfect squares to deduce, with justification, what the exponents of $2$ and $3$ must be in the factorization of $N$.
2. b) Reconstruct the number $N$, calculate its value, and write the complete list of its $9$ divisors to verify that it satisfies all the conditions.

The public bike rental service in an Irish city identifies each vehicle using a four-digit code with the structure $4X6Y$, where $X$ and $Y$ are unknown digits.

For maintenance reasons:

• The bikes undergo a monthly workshop check-up if their code is divisible by $8$ (the last three digits form a multiple of $8$).

• In addition, they enter the special weekly disinfection shift if their code is divisible by $11$.

1. a) Apply the divisibility rule for $11$ to find what value the sum of the digits $X + Y$ must have.
2. b) Taking into account that to be divisible by $8$ the digit $Y$ must be even, analyze the possible pairs of digits $(X, Y)$ and find the only code $4X6Y$ that allows the bike to undergo both the monthly check-up and the weekly disinfection.

A famous influencer wants to protect access to their video editing account with a four-digit PIN code of the form $5X4Y$. So as not to forget it, they have memorised the following conditions based on divisibility:

• The number is divisible by $4$.

• The number is also divisible by $9$.

• The units digit ($Y$) is greater than the thousands digit.

Find all possible PIN codes that satisfy these conditions.

On the main stage of a music festival, there are three robotic lighting systems. The first system emits a white flash every $18$ seconds, the second a blue flash every $24$ seconds, and the third an orange flash every $30$ seconds.

At $21:00$ hours, the three systems coincide for the first time. How many times will all three flashes coincide again before the concert ends at $22:25$ hours?

A logistics company needs to organise three types of aid packages in shipping containers. They have $144$ boxes of blankets, $120$ boxes of non-perishable food, and $84$ boxes of hygiene kits.

They want to load the maximum number of identical containers so that each container has the same number of boxes of each type and no boxes are left over in the warehouse.

1. How many containers can be sent?

2. How many boxes of each type will be in each container?

A natural number $N$ has the following prime factorisation: $N = 2^{a} \times 3^{2} \times 5^{1}$.

We know that the total number of factors of $N$ is exactly $24$.

1. Calculate the value of the exponent $a$.

2. Determine the numerical value of $N$.

3. Write down all the factors of $N$ that are multiples of $15$ but not of $2$.

A local council wants to pave a rectangular square measuring $45$ metres long by $24$ metres wide using square stone tiles. The architect wants the tiles to be as large as possible without having to cut any.

1. What should the side length of each tile be?

2. If due to budget constraints they decide to use tiles half that size, how many tiles will they now need to cover the entire square?

The mechanism of an old clock uses two cogs that mesh together. Cog $A$ has $54$ teeth and cog $B$ has $36$ teeth. If we place a red mark at the point where the teeth touch right now:

1. How many teeth of cog $A$ must pass before the marks coincide again for the first time?

2. How many turns will each cog have completed at that moment?

It is known that a number $X$ is a multiple of $12$ and $15$ simultaneously. Furthermore, $X$ has exactly $12$ natural divisors.

1. Using prime factorisation, deduce the value of $X$. (Hint: reason about the exponents of the factorisation).

2. Justify whether the number $X+1$ is a prime or composite number.

Three artificial satellites orbit the Earth. The first completes an orbit every $90$ minutes, the second every $2$ hours, and the third every $105$ minutes. If at $12:00$ noon they are aligned over the Greenwich meridian:

1. At what exact time will they be aligned again for the first time?

2. How many orbits will the slowest satellite have completed during that time?

To find the smallest positive integer $k$ by which $N = 2^4 \times 3^2 \times 5 \times 7^3$ must be multiplied so that the result is a perfect square, two students propose different approaches:

• Aoife's method: Multiplies all the factors with a calculator to obtain the full numerical value of $N$ and then tries multiplying by $2, 3, 4\dots$ until checking with the square root key whether the result is an integer.

• Seán's method: Directly examines the parity of the exponents in the prime factorization of $N$ to complete those that are not even.

1. a) Compare both methods and justify which one is more appropriate and generalizable, explaining the property of prime factorization that defines perfect squares.
2. b) Apply the more efficient method to calculate the exact value of $k$ and write the prime factorization of the new number $N \times k$.
3. c) Justify, using the formula for the total number of divisors, why the number $N \times k$ will necessarily have an odd number of divisors.

The railway company Irish Rail uses an anti-fraud system on its high-speed train tickets based on a six-digit check code with the format $2A4.5B2$, where $A$ and $B$ are unknown digits.

For the turnstile reader to validate the ticket, the code must satisfy three conditions simultaneously:

1. The number must be divisible by $6$.

2. The number must be divisible by $11$.

3. The digit $A$ must be strictly less than the digit $B$ ($A < B$).

1. a) Translate the divisibility rule for $6$ into a condition on the sum of the digits and find the possible values that the sum $A + B$ can take.
2. b) Apply the divisibility rule for $11$ (difference between the sum of the digits in odd positions and the sum in even positions) to deduce what exact mathematical relationship exists between $A$ and $B$.
3. c) Combine all the previous conclusions along with the condition $A < B$ to determine, with justification, all the valid ticket codes that the machine can generate.