Question

8. Suppose that a store offers gift certificates in denomina- tions of 25 dollars and 40 dollars. Determine the possible total amounts you can form using these gift certificates Prove your answer using strong induction

Answer 1

oid e povea dor, dis Solution fers ceficates of $25 Given that, store $40 $25 and $40 9044 5 and hence total are multiples of Pt is obvious that we cannoF amount that is not a muliple of values give 5 fo assume 5n dollars Let us Bor every ie. increase time we bave to 5 to get total toj 40/50 8 by 25/55 and possible amounts Possible cases assume) 5,8, t0, 13, 15, 18,20, 23, 25, 28, 30; 16, 21, 26,31, 32 Suvd $123 Gotat amoun HC) $40 25 2.5 40 1 50 2 65 13 75 2 2 90 100 4 O IO5 2 115 1 120 S 125 2. 130 1나0 IA5 150 IS5 2 160

all combinations, obsening By all We ane 11 obvious that gaps impossible to Ore Tt is clear that, we can form total amounts form from table, t135 o largest multiple 5 can't be achieved fining THceutificates Let PCO) be a statement that form asing s and $40. 5n dollars in we Can from above combinations & table PCn) is true n-28, 29, 30,31,32 Let K>32 i628jK then PGi) holds true to prove PCK) is true, we need to prove PCK+ is true we knouo K32 as KE32 -4 = 28 & P C28) is true

PCK-4) s true now add 25 ceutificate to it PCK-4)+ 25 is true pCk) = nlk) = 5 (K) knouw we as 5(K-4) + $25 is true 5 (K-4+5) is treie 50K+) is rue P(K+) i6 tvue PCK) is true K32 PC is true n28 Hence proved by foduction