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 $60$.

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 $B$.

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.

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.

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

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 $2$.