Question

DIVISIBLEBYNINE (x) 1 if x< 10 then if x 0 or x 9 then return "yes else return "no" 5 els while x> 0 do y y (x mod 10) Lx/10 X return DIVISIBLEBYNINE (y)

a. Argue that the algorithm always terminates, that is, that the algorithm does not call itself

recursively until all eternity. (Hint: Prove that the value of y in Line 9 is less than the value of

x at the beginning of the procedure and draw the right conclusions.)

b. Prove that the algorithm always gives the correct answer. (Hint: Use induction. For the

inductive step, argue that y is divisible by 9 if and only if this is true for x. Use the fact that 9

divides 10^k − 1 for any integer k.)

Answer 1

o) hebus prove bh.s by endoction on χ OH X410 olgowilhn olovtiousl tenminates becaupe i Creutes one to lines and, uhich take Consbant time 애1210, we proue that value of in line ends Contont timein line 1, 5-3 Gn Spe then voke bel on in line R noducion hupothes's necoive call on minale hu Coment inuota lion of eminat hen i :o Here lueny tminon-negalive betause Sinte χ210/they musl be iso,

2, s o and 10 b) Wet us pHove bhit by ind ion bnwe becoude o and aye only two numbes less than o and disible b we can aeo ume indclively tha't chvisitle by if y is dru isible by 9, then y:9J milaH e cris ble by 9ond

2 s not chulisble t, Lien whete yand tr t 1 olbo leave manywhn druided by druisible by9 and no anwe hetonned T is ogan ct On に。 lo'- is dhuistble by fon any inte gert churisible