Let-us-define-the-positive-number-n-with-four-digits-a-b-c-and-d-such-that-n-abcd-with-a-b-c-d-Z-1-a-9-0-b-9-0-c-9-and-0-d-9-Let-us-then-say-that-a-cool-number-is-a-four-digit-number-say-n-such-

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Question Number 4540 by Yozzii last updated on 06/Feb/16

$$Letusdefinethepositivenumbernwithfour$$$$digits\mathbf{a},\mathbf{b},\mathbf{c}and\mathbf{d}suchthatn=\mathbf{abcd}$$$$with\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}\in \mathbb{Z},1⩽\mathbf{a}⩽9,0⩽\mathbf{b}⩽9,$$$$0⩽\mathbf{c}⩽9and0⩽\mathbf{d}⩽9.Letusthensay$$$$thatacoolnumberisafourdigitnumber,$$$$sayn,suchthatthetwodigitnumberswrittenas$$$$\mathbf{ab}and\mathbf{cd}aregivenby\mathbf{ab}=r\times sand$$$$\mathbf{cd}=(r-1)\times (s+1)forsomenon-negativeintegers$$$$rands,r\ne s.Forexample,8081has$$$$\mathbf{a}=8,\mathbf{b}=0and80=10\times 8=while$$$$\mathbf{c}=8,\mathbf{d}=1and81=9\times 9=(10-1)(8+1).$$$$So,forn=8081,r=10whiles=8.$$$$Howmanyn,for1000⩽n⩽9999,arecool?$$$$$$$$Forn\in [1000,9999],n\in \mathbb{Z},howmanynexist$$$$sothat\mathbf{ab}=r\times sand\mathbf{cd}=r\times s+1?Call$$$$suchnwarmnumbers.$$$$$$

Commented by Rasheed Soomro last updated on 06/Feb/16

$$Forwarmnumber\mathbf{ab}=r\times sand\mathbf{c}\stackrel{?}{\mathbf{b}}=r\times s+1?$$$$Or\mathbf{ab}=r\times sand\mathbf{c}d=r\times s+1?Plconfirm.$$$$\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots$$$$Howmanynumbersarethere,whichare$$$$bothwarmandcoolatthesametime?$$$$Forexample$$$$484948=8\times 6\mathrm{\&}49=(8-1)(6+1)$$$$484948=8\times 6\mathrm{\&}49=8\times 6+1$$$$$$

Commented by Yozzii last updated on 06/Feb/16

$$Sorryforthetypo!{It}^{\prime }s\mathbf{cd}=\mathrm{r}\times \mathrm{s}+1.$$

Answered by Rasheed Soomro last updated on 08/Feb/16

$$CoolNumbers$$$${Let}^{\prime }sattacktheproblemfromr-sside.$$$$Wehavetofindnumberofpairs(r,s)$$$$withfollowingrestrictions:$$$${}^{\bullet }r\times smakesfirsttwodigitsfromleft\left(\mathbf{ab}\right)$$$$and\mathbf{a}⩾1,$$$$So,10⩽r\times s⩽99\dots \dots \dots \dots \dots \dots \dots \dots \dots .\left(i\right)$$$${}^{\bullet }(r-1)(s+1)makeslasttwodigits$$$$00⩽(r-1)(s+1)⩽99\dots \dots \dots \dots \dots ..\left(ii\right)$$$${}^{\bullet }randsarenon-negativeintegersandr\ne s$$$$butasr\times s\ne 0\left(from\left(i\right)\right),sor\ne 0\wedge s\ne 0,$$$$i-erandsarepositveintegers.$$$$r,s\in {\mathbb{Z}}^{+}withr\ne s\dots \dots \dots \dots \dots \dots \dots ..\left(iii\right)$$$$From\left(i\right)$$$$10⩽r\times s⩽99\Rightarrow \frac{10}{r}⩽s⩽\frac{99}{r}$$$$Butasr,s\in {\mathbb{Z}}^{+},So$$$$\lceil \frac{10}{r}\rceil ⩽s⩽\lfloor \frac{99}{r}\rfloor \dots \dots \dots \dots ..\left(iv\right)$$$$From\left(ii\right)$$$$00⩽(r-1)(s+1)⩽99\Rightarrow 0⩽s+1⩽\frac{99}{r-1}$$$$\Rightarrow -1⩽s⩽\frac{99}{r-1}-1$$$$\Rightarrow -1⩽s⩽\frac{100-r}{r-1}$$$$Butsincer,s\in {\mathbb{Z}}^{+},So$$$$-1⩽s⩽\lfloor \frac{100-r}{r-1}\rfloor \dots \dots \dots \dots .\left(v\right)$$$$From\left(iv\right)\mathrm{\&}\left(v\right)$$$$Max[\lceil \frac{10}{r}\rceil ,-1]⩽s⩽Min[\lfloor \frac{99}{r}\rfloor ,\lfloor \frac{100-r}{r-1}\rfloor ]$$$$Sincer>0\frac{10}{r}>-1,Max[\lceil \frac{10}{r}\rceil ,-1]=\lceil \frac{10}{r}\rceil$$$$\lceil \frac{10}{r}\rceil ⩽s⩽Min[\lfloor \frac{99}{r}\rfloor ,\lfloor \frac{100-r}{r-1}\rfloor ]\dots .\left(vi\right)$$$$s\ne r\dots \dots \dots \dots \dots \dots .\dots .from\left(iii\right)\dots ..\left(vii\right)$$$$From\left(vi\right)\mathrm{\&}\left(vii\right)$$$$\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots .$$$$\lceil \frac{10}{r}\rceil ⩽s⩽Min[\lfloor \frac{99}{r}\rfloor ,\lfloor \frac{100-r}{r-1}\rfloor ]\wedge s\ne r\dots \left(ix\right)$$$$\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots .$$$$Aboveconditiondeterminess,ifwefixr.$$$$Forr=4,sisgivenby$$$$\lceil \frac{10}{4}\rceil ⩽s⩽Min[\lfloor \frac{99}{4}\rfloor ,\lfloor \frac{100-4}{4-1}\rfloor ]$$$$\lceil 2.5\rceil ⩽s⩽Min[\lfloor 24.75\rfloor ,\lfloor \frac{96}{3}\rfloor ]$$$$3⩽s⩽Min[24,32]$$$$3⩽s⩽24\wedge s\ne 4$$$$s=3,5,6,7,\dots 24Total21values$$$$\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots$$$$r=1;s=10,11,\dots ..99\mid r=31;s=1,2$$$$r=2;s=5,6,\dots ,49\mid r=32;s=1,2$$$$r=3;s=4,5,\dots 33\mid r=33;s=1,2$$$$r=4;s=3,\stackrel{\times }{4},\dots 24\mid r=34;s=1,2$$$$r=5;s=2,3,4,\stackrel{\times }{5},\dots 19\mid r=35;s=1$$$$r=6;s=2,3,4,5,\stackrel{\times }{6},\dots 16\mid r=36;s=1$$$$r=7;s=2,3,..\stackrel{\times }{7},8,\dots 14\mid r=37;s=1$$$$r=8;s=2,3,\dots \stackrel{\times }{8},\dots 12(38,1),(39,1)\dots (50,1)\mid$$$$r=9;s=2,3,\dots .\stackrel{\times }{9},10,11$$$$r=10;s=1,2,\dots 9$$$$r=11;s=1,2,\dots 8$$$$r=12;s=1,2,\dots 8$$$$r=13;s=1,2,\dots 7$$$$r=14;s=1,2,\dots 6$$$$r=15;s=1,2,..,6$$$$r=16;s=1,2\dots 5$$$$r=17;s=1,2,\dots 5$$$$Outofspace$$

Commented by Yozzii last updated on 07/Feb/16

$${Isn}^{\prime }tit10⩽rs⩽99and0⩽(r-1)(s+1)⩽99?$$

Commented by Yozzii last updated on 07/Feb/16

$$Since\mathbf{ab}=\mathrm{r}\times \mathrm{s}\mathrm{and}\mathbf{cd}=(\mathrm{r}-1)(\mathrm{s}+1)$$$$\Rightarrow \mathbf{cd}=\mathrm{r}\times \mathrm{s}+\mathrm{r}-\mathrm{s}-1=\mathbf{ab}+\mathrm{r}-\mathrm{s}-1$$$$\Rightarrow \mathbf{cd}-\mathbf{ab}=\mathrm{r}-\mathrm{s}-1$$$$10⩽\mathbf{ab}⩽99and00⩽\mathbf{cd}⩽99$$$$min(\mathbf{cd}-\mathbf{ab})=min\left(\mathbf{cd}\right)-min\left(\mathbf{ab}\right)$$$$=00-10$$$$=-10$$$$max(\mathbf{cd}-\mathbf{ab})=max\left(\mathbf{cd}\right)-min\left(\mathbf{ab}\right)$$$$=99-10$$$$=89$$$$\Rightarrow 00-10⩽\mathbf{cd}-\mathbf{ab}⩽99-10$$$$-10⩽\mathrm{r}-\mathrm{s}-1⩽89$$$$-9⩽\mathrm{r}-\mathrm{s}⩽90$$$$\mathrm{s}-9⩽r⩽90+s$$$$Fromyouwehavealso\frac{10}{s}⩽r⩽\frac{99}{s}.$$$$Couldweplottheseinequalities$$$$togetthecommonregion?$$

Commented by Yozzii last updated on 07/Feb/16

Commented by Yozzii last updated on 07/Feb/16

Commented by Rasheed Soomro last updated on 07/Feb/16

$$Thankstomentionmistake,Ihavecorrectedit.$$

Commented by Rasheed Soomro last updated on 07/Feb/16

$$\mathcal{NICE}\mathcal{APPROACH}!$$

Commented by Rasheed Soomro last updated on 07/Feb/16

$$Pldoitmanuallyalso.Imeanwithout$$$$thehelpofWolframAlpha.$$

Commented by Yozzii last updated on 07/Feb/16

$$ThebestthingIknowtodo$$$$isdrawthegraphsmanuallyandcount$$$$thevalidintegerpairs\dots$$$$Ilackexperienceinsolvingsuch$$$$inequalitiesviaashortermethod.$$

Commented by Rasheed Soomro last updated on 08/Feb/16

$$Thanksforreply.$$

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