Solve part D . Show a deatiled proof, explain all assumptions, theorems, lemma,

Question

Solve part D. Show a deatiled proof, explain all assumptions, theorems, lemma, definitions being used to solve problem.

d. Let a,be Nt he coprime, Call an c N purchasable if the equation has a solution with r,yN. Let P denote the set of purchasable numbers (a) Show for all c ? Z there exists a unique solution to ax + by-c with x ? z and y ? [o,...,a -1. Denote this solution by (xe, ye). (2 pts) b) Show that for cEN one has that c P if and only if e E N. (4 pts) (c) Deduce that if c > ab- a b then c is purchasable (Hint: use (b) to parameterize the set of non-purchasble c and find the maximum of this set). (2 pts) (d) The (insert favorite thing here) store is having a sale on (insert favorite thing here)'s, but they are only being sold in groups of 6 and 10. Which quantities of (insert favorite thing here) can you buy? (1 pts)

Explanation / Answer

Since from part (c) we know that the quantity c is purchasable if c > ab-a-b and here they are being sold in group of 6 and 10 so if we consider a=6 and b= 10 then ab-a-b= 60-6-10=44 so we can buy the quantities of (say X) which are greater than 44 means all quantities of the form 6x+10y> 44 we can buy.

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