On planet Enigma, the residents use a currency called the confusion. There are only 2 confusion bills on Enigma, one worth 8 confusions and

b )   8x + 11y = 97      ....I'm with Melody, here......"trial and error" might be the shortest distance between two points, since the numbers are small

Note that  x can't be greater than 12 and y can't be greater than 8 ....  and y must be odd because 8x is always even.....and the only way to get an odd (97, in this case) is to add an odd to an even....

When y = 3, x = 8

And y can't be 1, 5 or 7

So.... 8 "8" confusions and 3 "11" confusions seem to work.......

Hi Mellie,

Now, we can do this the easy way .... or we can do it the hard way.

I'll start with the easy way

11x+8y=96

if x=0     8y=96    y=12          So   12 * 8 confusion notes will be good

if x=1    

11+8y=96

8y=85     No integer solution

if x=2          22+8y=96        8y=74        No integer solution

if x=3           33+8y=96        no integer solution

Just keep doing this and you will find that the only other solution is     

8*11confusions+1*8confusion      88+8=96

So there are only 2 ways.

You can do the other two similarly. :)    

This method is time consuming but the numbers are all quite small so it is quite doable :)

The other way is to solve the diophantine equation more formally.

I have done the first one on a peice of paper in front of me but it takes even longer to get to your answer than the easier way that I just used, so do it the easy way :))

A: I can only come up with two:    Twelve   8  Confusions bills

                                               One 8 Confusions bill and eight 11 Confusions bills

Take 96 and remove multiples of 11 from it succsessively    96  85  74  63  52  41  30  19  8     only one of which is divisible by 8......That is the only combo of 11s and 8s that will work...

Likewise with the 8s   :  96 88 80 72 64 56 48  40  32 24  16  8    only 88 is divisible by 11 .....

B:  Do the same thing for 97 :  97  86  75  64  53  42  31  20  9  1       I find one way of buying something for 97 confusions with only 8s and 11s....    eight 8s  and three 11s

C.  For 69     69  58  47  36  25  14  3  (nothing divisible by 8)   I find no way to pay 69 using only 8s and 11s

                   69  61  53  45  37  29  21  13  5   (nothing evenly divisible by 11)       I think you will get some 10kg bitcoins!!

Thanks Chris and anon - I think we might need change too :((

Mellie this isn't really for you - I am just practicing doing diophantine equations  :))

Part C

8A+11B=69

The FCF of 8 and 11 is 1    and 1 goes into 69 so this does have integral solutions  :/

8*-276+11*207=69

8A+11B=69

So 

A=-276+11N          B=207-8N

276/11=25 and a bit

A=-276+11*26=10         B=207-8*26=-1

Mmm       There are not any solutions!

More practice

Part B

11A+8B=94

Largest common factor of 11 and 8 is 1.  One is a factor of 94.  So this does have integral solution.

2 elevens and 9 eights confusions is the only solution.