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Question 1
$15$ has factors $3$ and $5$ (and $1$ and $15$). Write out all the numbers between $1$ and $14$ inclusive and then cross out everything which is a multiple of $3$, or $5$. You should be left with $8$ numbers which are co-prime with $15$.
Question 2
For example, $\phi(7)=6$ as $1, 2, 3, 4, 5$ and $6$ share no common factors with $7$. If one of the first $6$ numbers did share a factor with $7$, then $7$ would not be prime.
Question 3
For example $\phi(3^2)= 6$ since $1, 2,$ _ $, 4, 5,$ _ $, 7, 8$ are coprime with $9$.
Try a few more examples for different prime numbers and different powers before trying to generalise.
Question 4
$24$ can be written as $3 \times 8$. Is it true that $\phi(24) = \phi(3) \times \phi(8)$?
Alternatively $24$ can be written as $4 \times 6$. Is it true that $\phi(24) = \phi(4) \times \phi(6)$?
Question 5
It might be helpful to write $n$ as a product of prime factors, e.g. $n=p_1^{a_1} \times p_2^{a_2} \times \cdots \times p_k^{a_k}$.
This problem asks you to use your curve sketching knowledge to find all the solutions to an equation.
Can you find a three digit number which is equal to the sum of the hundreds digit, the square of the tens digit and the cube of the units digit?
Frosty the Snowman is melting. Can you use your knowledge of differential equations to find out how his volume changes as he shrinks?