Question

Exercise 8.6.1: Proofs by strong induction - combining stamps. Prove each of the following statements using strong induction Prove that any amount of postage worth 8 cents or more can be made from 3-cent or 5-cent stamps. (0) Prove that any amount of postage worth 24 cents or more can be made from 7-cent or 5-cent stamps. Prove that any amount of postage worth 12 cents or more can be made from 3-cent or 7-cent stamps

Answer 1

We prove result by mathematical induction We prove amount of postage worth 8 cents or more can be made from 3 cents ar scents stamps let. nos. we show na 30+ 5b , auben usoy Step O nas Then choose aclobal s na 3+5=8 a. Result hold for nag naq, choose a 3, ball Ә) n= 3x3 =g and 9 cents stamps can made with made from a lents three stamps na 10 . choose aco, b=2 .) na 5x2 alo . Result hold for nalo . Step @ : Assume result hold for n=j je . ja 39, +52afor some qu, bye N uPo} step ② We prove result for najti Now it la Bart5b, t1 = 30, 456, +6-5. = 3(9,+2) +506-12 j+1 = 3a'+56! ,ai, ble Nexos Result hold for jal . By induction bypothesis result hold for all n>8 non

of step ® : We prove result for j+/ Now ja Bertsb, Also result hold for all niej (o! by hypothesis) for j-2, we have giza 392 + 5bg for some az, be fin Oto} Add one z cents stamp in 3+2+3 - joti a 394.+562 73 j+1 = 3 Cytl) +5b2 jtla zal+5bą, al, be FW O{0} a result hold for najti . By induction by po thesis result hold for all D 28. nein u{Oy We prove postage stamp worth 24 cents or more can be made from a cent or 5 cents stamp - ie for ny, 24 we show na 5art a,b fin u 203 step 0 1224 choose 922, baz : 24 582+7x2 " Result hold for na24 9225 choose ans, bao a 25=545 Result hold for n=25 . ho a6 choose act. baz -' 26 5W) + 7*3 =result Se hord

na af choose a-4, bal > 5* G * F*1527 na 28 choose aco, b=4 => 7x4=28 " Result hold for. n.28 Step : Assume result hold for maj, j = 29 i j = 59, + fb,, a, b, ein U20 Step ③ - love prone result for najti consider j+1 = 57,475,+1 59 +76 +21 - 20 56a-4) + 706+7) stl = 597762 ;dz , bz F AN UXo? . Result hold for jal As result hold for neg. J 7, 29 s's for 3-4 = 59'+ t6! , a b' FIN ► j-4+5= +1 = 50'+76'+5 = 5 cartel + 76! St1 = 59, ttba azoba fpN ulog . : Result hold for najti . By induction hypothesis Result hold for all ny, 24 norn.u{0}

③ We prove amount of postage worth 12 cents or more can be made from 3 cents or q cents stamp let A712; we show na 39476, albo Nuloy Step © :- n512 choose a=4, bco → 488=12 .. Result hold for n=12. nalg Choose 952, bal a 3xR4 7x15 13 Result hold for nale step step n=14, choose aco, b:a . 7* 2014 . Result hold for n=14 :- Assume result hold for naj' .1215 le ja 39, + 76 ,a,ubi en : We prove result for nefti non As result hold for nej a as for -2 - j-2 = 30.4 76' 333-2+3 ajtla 3a't76'+3 =3 cale) 7L! st = 39 *Ibe ali bo finulog si Results hold for jties . By induction hypothesis, nesik hold for all oxie