1. Susie thinks of a positive integer $n$. She notices that, when she divides 2023 by $n$, she is left with a remainder of 43 . Find how many possible values of $n$ there are.
Solution
Since Susie's number leaves a remainder of 43 when dividing it into 2023, we know that
$$
2023=k n+43
$$
where $k$ is an integer. Also, note that $n>43$, because the remainder needs to be smaller than the number Susie divides by.

Hence $k n=1980$, so Susie's number could be any factor of 1980 which is larger than 43 .
$$
1980=2^{2} \times 3^{2} \times 5 \times 11 \text {. }
$$

Now we could just write a list of all the factors of 1980 , but it's a little quicker to recognise that, aside from $44 \times 45$, each pair of factors of 1980 will contribute exactly one possible value of $n$.

We can count the total number of factors by product of the numbers one higher than each of the powers of the prime factorisation. The total number of factors of 1980 is $(2+1)(2+1)(1+$ 1) $(1+1)=36$. In each pair apart from 44 and 45 , one is above 43 . This leads to the total number of possible values of $n$ being 19 .

For reference although not required by the question, those numbers are 1980, 990, 660, 495, $396,330,220,198,180,165,132,110,99,90,66,60,55,45$ and 44.