# Assignment 10, STEP question i.b

I've got two things here. One about the assignment, the other about STEP preparation in general.

First, I've done a thing which to me seems to work, but I don't know if it qualifies as an answer. It's not one which is listed, but still.

Basically, I've said that p1 * p2 * p3 * ... * pk will always be an integer factor of N. When we multiply N by the fractions, we don't really need to multiply out the brackets (so we can just keep the numerator as N(p1 - 1)(p2 - 1)...(pk - 1)). We could then write out the whole thing as (N/p1p2p3...pk)[(p1 - 1)(p2 - 1)...(pk - 1)]. N/p1p2p3...pk will be an integer, since the numerator was already stated to be an integer and the denominator is an integer factor, as shown prior.
(p1 - 1)(p2 - 1)...(pk - 1) will also always be an integer, since each bracket is an integer (as pk is an integer and so is 1). So now we have two integers: (N/p1p2p3...pk) and [(p1 - 1)(p2 - 1)...(pk - 1)]. Multiplying two integers together will always result in an integer, so we have proved that f(N) will always be an integer for any positive integer N.

Does this work?

Okay, second issue; I'm getting the feeling that I'm not preparing enough. Basically, I've done 10 of the foundation assignments (including this one). Which of those have I completed without any hints? 2. Assignment 4 and assignment 6. Note that this is using the hints in general, not necessarily the STEP questions.
But still, is this sort of thing bad? (I've already made a topic regarding this, but I've just realized how many times I've actually used hints, whereas before I was just feeling bad about using them once or twice).
If it is bad, then what other things can I do alongside these foundation courses? I don't want to be using them as much as I am, by a long shot.

### For your first question, your

For your first question, your solution seems mostly well explained. It might be helpful to say that $N=p_1^{r_1}p_2^{r_2}\ldots p_k^{r_k}$ where $r_i \geq 1$ for $i=1,2,\ldots, k$ which means $p_1 p_2 \ldots p_k$ divides $N$.

With regards to STEP preparation in general, keep at it as it does get easier the more you practise. If you're worried about overusing the hints, perhaps set yourself a target to only refer to them a limited number of times for the next few assignments you attempt?

### The hints are there to be used!

To comment on your second issue, please do not be concerned about using the hints a lot for the foundation assignments - that's what they are there for! They are designed to get you into the mindset of answering STEP questions. You still have a long time to go before the exams - some students don't even begin preparing for STEP until October or even January so you have a head start. If you are still relying on lots of hints next April, that would be the time to worry!

A good way to use this programme of STEP support is to look at material without hints, then after a good period of thinking time, and perhaps going away and coming back to a question, look at the hints. This is designed to help you develop the skill of essentially thinking of your own hints, until you can solve many STEP questions unaided.

Underground Mathematics: Selected worked STEP questions

STEP Question database

University of Cambridge Mathematics Faculty: What do we look for?

University of Cambridge Mathematics Faculty: Information about STEP