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Q4.

If you divide both by n, then the question becomes when is (n-1)! divisible by n.

If n is any prime this cannot happen as none of the factorial factors can contain it because it's Prime and these are all smaller.

So what if n is not Prime?  It must have a decomposition into factors all of which are smaller and so contained somewhere in the factors of the factorial.  But what if such a factor occurs twice?  It can be further factored using numbers we haven't yet used so n will divide the factorial.

This sounds a bit clumsy so I'll illustrate with an example.

Suppose n = 147 = 3 x 7squared

146! = 3 x 7 x 14 x other numbers so 147 divides it.

So all composite numbers have the property.

Bob

hi thehay95

Welcome to the forum.

I've had a go at sketching this solid.  Trouble is that just having the letters PQRSTUV doesn't exactly tell me which points to join to which.  It looks like there's a prism PQRSTV, which has a triangular cross section QRV, and sitting on this is a rectangular based pyramid TVRSU, where the base is TVRS and the vertex U.

Volume of the prism should be fairly straight forward.  Calculate the area of the triangle QRV and multiply by the length, PQ.

The pyramid is more tricky.  You'll need the area of the base, TVRS.  This is a rectangle.  You need Pythag. to calculate VR.  Then the perpendicular height from TVRS to U.  If you draw the triangle UZY, you'll see that UZ is not perpendicular to ZY so that length won't do.  But you can calculate UY and the angle UYZ and so treating UY as a hypotenuse you can get the required height by opposite (height) = hyp x sin(UYZ).  Once you have this the volume of the pyramid is  one third of the area of base times the perpendicular height.

Then just add together the two volumes.  :)

Hope that helps,

image upload problem.  I have a diagram but I cannot log in to imgur to upload it.  Some problem with a Vodafone security certificate ??? I'm on BT broadband.  Anyone got any suggestions.  It happens with both IE11 and Chrome. My account still exists as old images are still showing on the forum

Bob

hi joseph

Welcome to the forum.

Thanks for the supportive comments.  MIF is funded by carefully selected advertisers.  Please don't use the forum for free advertising.

Bob

hi Mathegocart

Typos can creep into questions.  In the UK the national exams are, nowadays, subject to a load of checking but question errors still get in sometimes.  [eg. https://www.independent.co.uk/news/educ … 27141.html ]

Or maybe it was intended to test if you are aware that sometimes a calculation cannot be completed.  If I was going to do this I would word my questions (have to be several) "Work out all the missing side lengths and angles in these triangles.  If you think a triangle cannot exist explain why."  Something like that would be reasonable if it was included like that as part of the assessment.  If it was a single question on the topic, and you have copied it correctly**, then it looks like a serious error.  What was the test?

Bob

** If the numbers are swapped around then it is possible for example.

hi Mathegocart

No error.  No triangle can be made with those measurements.  Try it yourself.  Draw AB = 27 (may need to scale down) and angle B = 38.  You don't know how long to make BC so just continue the line for some distance.  Set a compass to a radius of AC = 15 and try to intersect BC.  This fails.  The shortest distance from A to BC is about 18.

Bob