Question-184477

Question Number 184477 by Shrinava last updated on 07/Jan/23

Commented by JDamian last updated on 07/Jan/23

I don't even understand your question. Would you mind express in math language?

Commented by mr W last updated on 07/Jan/23

when C(n)=core of number n, then C(21)=18 C(12)=9

$${when}\:{C}\left({n}\right)={core}\:{of}\:{number}\:{n},\:{then} \\ $$$${C}\left(\mathrm{21}\right)=\mathrm{18} \\ $$$${C}\left(\mathrm{12}\right)=\mathrm{9} \\ $$

Answered by Rasheed.Sindhi last updated on 07/Jan/23

•tu^(−) is 2-digit number.Clearly u∈{0,1,2,...,9} & t∈{1,2,3,...,9} •Core of tu^(−) is either 9 or 18 and this depends upon values of u and t as follows: determinant ((u,(values of t_(for core 9_(/total) ) ),(values of t_(for core 18_(/total) ) )),(0,(1≤t≤9 9),(− 0)),(1,(− 0),(1≤t≤9 9)),(2,(t=1 1),(2≤t≤9 8)),(3,(t=1,2 2),(3≤t≤9 7)),(4,(1≤t≤3 3),(4≤t≤9 6)),(5,(1≤t≤4 4),(5≤t≤9 5)),(6,(1≤t≤5 5),(6≤t≤9 4)),(7,(1≤t≤6 6),(7≤t≤9 3)),(8,(1≤t≤7 7),(t=8,9 2)),(9,(1≤t≤8 8 ),(t=9 1)),(,(TOTAL 45),( 45))) Sum of the cores=9×45+18×45 =1215

$$\bullet\overline {{tu}}\:{is}\:\mathrm{2}-{digit}\:{number}.{Clearly} \\ $$$$\:\:{u}\in\left\{\mathrm{0},\mathrm{1},\mathrm{2},…,\mathrm{9}\right\}\:\&\:{t}\in\left\{\mathrm{1},\mathrm{2},\mathrm{3},…,\mathrm{9}\right\} \\ $$$$\bullet{Core}\:{of}\:\overline {{tu}}\:{is}\:{either}\:\mathrm{9}\:{or}\:\mathrm{18}\: \\ $$$$\:\:\:{and}\:\:{this}\:{depends}\:{upon}\:{values}\:{of} \\ $$$$\:\:\:{u}\:{and}\:{t}\:{as}\:{follows}:\: \\ $$$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|}{{u}}&\hline{\underset{\underset{/{total}} {{for}\:{core}\:\mathrm{9}}} {{values}\:{of}\:{t}}}&\hline{\underset{\underset{/{total}} {{for}\:{core}\:\mathrm{18}}} {{values}\:{of}\:{t}}}\\{\mathrm{0}}&\hline{\mathrm{1}\leqslant{t}\leqslant\mathrm{9}\:\:\:\:\:\:\:\mathrm{9}}&\hline{−\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{0}}\\{\mathrm{1}}&\hline{−\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{0}}&\hline{\mathrm{1}\leqslant{t}\leqslant\mathrm{9}\:\:\:\:\:\:\:\:\:\:\mathrm{9}}\\{\mathrm{2}}&\hline{{t}=\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}}&\hline{\mathrm{2}\leqslant{t}\leqslant\mathrm{9}\:\:\:\:\:\:\:\:\:\:\mathrm{8}}\\{\mathrm{3}}&\hline{{t}=\mathrm{1},\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}}&\hline{\mathrm{3}\leqslant{t}\leqslant\mathrm{9}\:\:\:\:\:\:\:\:\:\:\mathrm{7}}\\{\mathrm{4}}&\hline{\mathrm{1}\leqslant{t}\leqslant\mathrm{3}\:\:\:\:\:\:\:\:\mathrm{3}}&\hline{\mathrm{4}\leqslant{t}\leqslant\mathrm{9}\:\:\:\:\:\:\:\:\:\:\mathrm{6}}\\{\mathrm{5}}&\hline{\mathrm{1}\leqslant{t}\leqslant\mathrm{4}\:\:\:\:\:\:\:\:\mathrm{4}}&\hline{\mathrm{5}\leqslant{t}\leqslant\mathrm{9}\:\:\:\:\:\:\:\:\:\:\mathrm{5}}\\{\mathrm{6}}&\hline{\mathrm{1}\leqslant{t}\leqslant\mathrm{5}\:\:\:\:\:\:\:\:\mathrm{5}}&\hline{\mathrm{6}\leqslant{t}\leqslant\mathrm{9}\:\:\:\:\:\:\:\:\:\:\mathrm{4}}\\{\mathrm{7}}&\hline{\mathrm{1}\leqslant{t}\leqslant\mathrm{6}\:\:\:\:\:\:\:\:\mathrm{6}}&\hline{\mathrm{7}\leqslant{t}\leqslant\mathrm{9}\:\:\:\:\:\:\:\:\:\:\mathrm{3}}\\{\mathrm{8}}&\hline{\mathrm{1}\leqslant{t}\leqslant\mathrm{7}\:\:\:\:\:\:\:\:\mathrm{7}}&\hline{{t}=\mathrm{8},\mathrm{9}\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}}\\{\mathrm{9}}&\hline{\mathrm{1}\leqslant{t}\leqslant\mathrm{8}\:\:\:\:\:\:\:\:\mathrm{8}\:\:\:\:\:\:\:\:}&\hline{{t}=\mathrm{9}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}}\\{}&\hline{{TOTAL}\:\:\:\:\:\mathrm{45}}&\hline{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{45}}\\\hline\end{array} \\ $$$${Sum}\:{of}\:{the}\:{cores}=\mathrm{9}×\mathrm{45}+\mathrm{18}×\mathrm{45} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{1215}\:\: \\ $$

Answered by Rasheed.Sindhi last updated on 08/Jan/23

AnOther way: •tu^(−) is 2-digit number.Clearly u∈{0,1,2,...,9} & t∈{1,2,3,...,9} Observations: •Number of 2-digit numbers =99−9=90 •Core of tu^(−) is either 9 or 18 •All numbers in which t<u have core 9, other numbers(in which t≥u but u≠0) have core 18 •In case u=0 core of t0^(−) is 9 although t>u • Number of the numbers having core 9 is 45.^★ So sum of cores is 9 × 45 = 405 • Obviously number of the remaining numbers(in which t≥u and having cores 18) is 90−45=45.So the sum of cores 18×45=810 •Sum of cores of all 2-digit number is therefore 405+810=1215 ^★ The way of counting is given in my other answer.The better way of counting may be suggested by sir mr W

$$\boldsymbol{\mathrm{AnOther}}\:\boldsymbol{\mathrm{way}}: \\ $$$$\bullet\overline {{tu}}\:{is}\:\mathrm{2}-{digit}\:{number}.{Clearly} \\ $$$$\:\:{u}\in\left\{\mathrm{0},\mathrm{1},\mathrm{2},…,\mathrm{9}\right\}\:\&\:{t}\in\left\{\mathrm{1},\mathrm{2},\mathrm{3},…,\mathrm{9}\right\} \\ $$$$\boldsymbol{{Observations}}: \\ $$$$\bullet{Number}\:{of}\:\mathrm{2}-{digit}\:{numbers} \\ $$$$\:\:\:\:=\mathrm{99}−\mathrm{9}=\mathrm{90} \\ $$$$\bullet{Core}\:{of}\:\overline {{tu}}\:{is}\:{either}\:\mathrm{9}\:{or}\:\mathrm{18} \\ $$$$\bullet\boldsymbol{{All}}\:\boldsymbol{{numbers}}\:\boldsymbol{{in}}\:\boldsymbol{{which}}\:\:\boldsymbol{{t}}<\boldsymbol{{u}}\:\:\boldsymbol{{have}} \\ $$$$\:\:\:\boldsymbol{{core}}\:\:\mathrm{9},\:\boldsymbol{{other}}\:\boldsymbol{{numbers}}\left(\boldsymbol{{in}}\:\boldsymbol{{which}}\right. \\ $$$$\left.\:\:\:\boldsymbol{{t}}\geqslant\boldsymbol{{u}}\:\boldsymbol{{but}}\:\boldsymbol{{u}}\neq\mathrm{0}\right)\:\boldsymbol{{have}}\:\boldsymbol{{core}}\:\mathrm{18} \\ $$$$\bullet\boldsymbol{{In}}\:\boldsymbol{{case}}\:\boldsymbol{{u}}=\mathrm{0}\:\boldsymbol{{core}}\:\boldsymbol{{of}}\:\:\overline {\boldsymbol{{t}}\mathrm{0}}\:\boldsymbol{{is}}\:\mathrm{9}\: \\ $$$$\:\:\:\boldsymbol{{although}}\:\boldsymbol{{t}}>\boldsymbol{{u}} \\ $$$$\bullet\:{Number}\:{of}\:{the}\:{numbers}\:{having} \\ $$$$\:\:\:\:{core}\:\mathrm{9}\:{is}\:\mathrm{45}.^{\bigstar} \: \\ $$$$\:\:\:{So}\:{sum}\:{of}\:{cores}\:{is}\:\:\mathrm{9}\:×\:\mathrm{45}\:=\:\mathrm{405} \\ $$$$\bullet\:{Obviously}\:{number}\:{of}\:{the}\:{remaining} \\ $$$$\:\:\:{numbers}\left({in}\:{which}\:{t}\geqslant{u}\:{and}\:{having}\right. \\ $$$$\left.\:\:\:{cores}\:\mathrm{18}\right)\:{is}\:\mathrm{90}−\mathrm{45}=\mathrm{45}.{So}\:{the}\:{sum} \\ $$$$\:\:\:{of}\:{cores}\:\mathrm{18}×\mathrm{45}=\mathrm{810} \\ $$$$\bullet{Sum}\:{of}\:{cores}\:{of}\:\:{all}\:\mathrm{2}-{digit}\:{number} \\ $$$$\:\:\:{is}\:\:{therefore}\:\:\mathrm{405}+\mathrm{810}=\mathrm{1215} \\ $$$$\:^{\bigstar} {The}\:{way}\:{of}\:{counting}\:{is}\:{given}\:{in} \\ $$$$\:\:\:\:{my}\:{other}\:{answer}.\boldsymbol{{The}}\:\boldsymbol{{better}}\:\boldsymbol{{way}} \\ $$$$\:\:\boldsymbol{{of}}\:\boldsymbol{{counting}}\:\:\boldsymbol{{may}}\:\boldsymbol{{be}}\:\boldsymbol{{suggested}} \\ $$$$\:\:\boldsymbol{{by}}\:\boldsymbol{{sir}}\:\boldsymbol{{mr}}\:\boldsymbol{{W}} \\ $$

Commented by mr W last updated on 08/Jan/23

i think you have greatly solved sir. numbers with t<u have the core 9, while numbers with t≥u have the core 18^(∗)) . this is just an observation. to be mathematically strict, it must be proved. ^(∗)) there are exceptions! e.g. the core of 20 is 9, not 18.

$${i}\:{think}\:{you}\:{have}\:{greatly}\:{solved}\:{sir}. \\ $$$$\boldsymbol{{numbers}}\:\boldsymbol{{with}}\:\boldsymbol{{t}}<\boldsymbol{{u}}\:\boldsymbol{{have}}\:\boldsymbol{{the}}\:\boldsymbol{{core}} \\ $$$$\mathrm{9},\:\boldsymbol{{while}}\:\boldsymbol{{numbers}}\:\boldsymbol{{with}}\:\boldsymbol{{t}}\geqslant\boldsymbol{{u}}\:\boldsymbol{{have}}\: \\ $$$$\boldsymbol{{the}}\:\boldsymbol{{core}}\:\mathrm{18}\:^{\left.\ast\right)} . \\ $$$${this}\:{is}\:{just}\:{an}\:{observation}.\:{to}\:{be} \\ $$$${mathematically}\:{strict},\:{it}\:{must}\:{be} \\ $$$${proved}. \\ $$$$ \\ $$$$\:^{\left.\ast\right)} \:{there}\:{are}\:{exceptions}!\:{e}.{g}.\:{the}\:{core} \\ $$$${of}\:\mathrm{20}\:{is}\:\mathrm{9},\:{not}\:\mathrm{18}. \\ $$

Commented by Rasheed.Sindhi last updated on 08/Jan/23

Grateful sir! I′ve added now the exception.Pl reread my answer. As far as the proof is concerned I know only one way that is to test the proposition for all numbers(10, 11,12,...99). determinant (((10),(20),(30),(40),(50),(60),(70),(80),(90)),((11),(21),(31),(41),(51),(61),(71),(81),(91)),((12),(22),(32),(42),(52),(62),(72),(82),(92)),((13),(23),(33),(43),(53),(63),(73),(83),(93)),((14),(24),(34),(44),(54),(64),(74),(84),(94)),((15),(25),(35),(45),(55),(65),(75),(85),(95)),((16),(26),(36),(46),(56),(66),(76),(86),(96)),((17),(27),(37),(47),(57),(67),(77),(87),(97)),((18),(28),(38),(48),(58),(68),(78),(88),(98)),((19),(29),(39),(49),(59),(69),(79),(89),(99))) Blue numbers have core 9 Red numbers have core 18

$$\boldsymbol{{Grateful}}\:\boldsymbol{{sir}}!\:{I}'{ve}\:{added}\:{now}\:{the} \\ $$$${exception}.{Pl}\:{reread}\:{my}\:{answer}. \\ $$$${As}\:{far}\:{as}\:{the}\:{proof}\:{is}\:{concerned}\:{I} \\ $$$${know}\:{only}\:{one}\:{way}\:{that}\:{is}\:{to}\:{test} \\ $$$${the}\:{proposition}\:{for}\:{all}\:{numbers}\left(\mathrm{10},\right. \\ $$$$\left.\mathrm{11},\mathrm{12},…\mathrm{99}\right). \\ $$$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|}{\mathrm{10}}&\hline{\mathrm{20}}&\hline{\mathrm{30}}&\hline{\mathrm{40}}&\hline{\mathrm{50}}&\hline{\mathrm{60}}&\hline{\mathrm{70}}&\hline{\mathrm{80}}&\hline{\mathrm{90}}\\{\mathrm{11}}&\hline{\mathrm{21}}&\hline{\mathrm{31}}&\hline{\mathrm{41}}&\hline{\mathrm{51}}&\hline{\mathrm{61}}&\hline{\mathrm{71}}&\hline{\mathrm{81}}&\hline{\mathrm{91}}\\{\mathrm{12}}&\hline{\mathrm{22}}&\hline{\mathrm{32}}&\hline{\mathrm{42}}&\hline{\mathrm{52}}&\hline{\mathrm{62}}&\hline{\mathrm{72}}&\hline{\mathrm{82}}&\hline{\mathrm{92}}\\{\mathrm{13}}&\hline{\mathrm{23}}&\hline{\mathrm{33}}&\hline{\mathrm{43}}&\hline{\mathrm{53}}&\hline{\mathrm{63}}&\hline{\mathrm{73}}&\hline{\mathrm{83}}&\hline{\mathrm{93}}\\{\mathrm{14}}&\hline{\mathrm{24}}&\hline{\mathrm{34}}&\hline{\mathrm{44}}&\hline{\mathrm{54}}&\hline{\mathrm{64}}&\hline{\mathrm{74}}&\hline{\mathrm{84}}&\hline{\mathrm{94}}\\{\mathrm{15}}&\hline{\mathrm{25}}&\hline{\mathrm{35}}&\hline{\mathrm{45}}&\hline{\mathrm{55}}&\hline{\mathrm{65}}&\hline{\mathrm{75}}&\hline{\mathrm{85}}&\hline{\mathrm{95}}\\{\mathrm{16}}&\hline{\mathrm{26}}&\hline{\mathrm{36}}&\hline{\mathrm{46}}&\hline{\mathrm{56}}&\hline{\mathrm{66}}&\hline{\mathrm{76}}&\hline{\mathrm{86}}&\hline{\mathrm{96}}\\{\mathrm{17}}&\hline{\mathrm{27}}&\hline{\mathrm{37}}&\hline{\mathrm{47}}&\hline{\mathrm{57}}&\hline{\mathrm{67}}&\hline{\mathrm{77}}&\hline{\mathrm{87}}&\hline{\mathrm{97}}\\{\mathrm{18}}&\hline{\mathrm{28}}&\hline{\mathrm{38}}&\hline{\mathrm{48}}&\hline{\mathrm{58}}&\hline{\mathrm{68}}&\hline{\mathrm{78}}&\hline{\mathrm{88}}&\hline{\mathrm{98}}\\{\mathrm{19}}&\hline{\mathrm{29}}&\hline{\mathrm{39}}&\hline{\mathrm{49}}&\hline{\mathrm{59}}&\hline{\mathrm{69}}&\hline{\mathrm{79}}&\hline{\mathrm{89}}&\hline{\mathrm{99}}\\\hline\end{array} \\ $$$$\mathcal{B}{lue}\:{numbers}\:{have}\:{core}\:\mathrm{9} \\ $$$${Red}\:\:{numbers}\:{have}\:{core}\:\mathrm{18} \\ $$

Commented by mr W last updated on 08/Jan/23

$${your}\:{solution}\:{is}\:{great}! \\ $$$${as}\:{for}\:{the}\:{proof},\:{see}\:{my}\:{attempt}. \\ $$

Answered by mr W last updated on 08/Jan/23

there are 3 typs of 2−digit numbers typ 1: like 10, 20 9 numbers, core of each is 9 typ 2: like 11, 22 9 numbers, core of each is 18 typ 3: like 12, 23 or 21, 32 9×8=72 such numbers. one half of them, like 12, have the core 9, the other half, like 21, have the core 18. the sum of the cores of all 2−digit numbers: 9×8+9×18+36×9+36×18=1215

$${there}\:{are}\:\mathrm{3}\:{typs}\:{of}\:\mathrm{2}−{digit}\:{numbers} \\ $$$${typ}\:\mathrm{1}:\:{like}\:\mathrm{10},\:\mathrm{20} \\ $$$$\mathrm{9}\:{numbers},\:{core}\:{of}\:{each}\:{is}\:\mathrm{9} \\ $$$${typ}\:\mathrm{2}:\:{like}\:\mathrm{11},\:\mathrm{22} \\ $$$$\mathrm{9}\:{numbers},\:{core}\:{of}\:{each}\:{is}\:\mathrm{18} \\ $$$${typ}\:\mathrm{3}:\:{like}\:\mathrm{12},\:\mathrm{23}\:{or}\:\mathrm{21},\:\mathrm{32} \\ $$$$\mathrm{9}×\mathrm{8}=\mathrm{72}\:{such}\:{numbers}.\:{one}\:{half}\:{of}\:{them}, \\ $$$${like}\:\mathrm{12},\:{have}\:{the}\:{core}\:\mathrm{9},\:{the}\:{other}\:{half}, \\ $$$${like}\:\mathrm{21},\:{have}\:{the}\:{core}\:\mathrm{18}. \\ $$$$ \\ $$$${the}\:{sum}\:{of}\:{the}\:{cores}\:{of}\:{all}\:\mathrm{2}−{digit} \\ $$$${numbers}: \\ $$$$\mathrm{9}×\mathrm{8}+\mathrm{9}×\mathrm{18}+\mathrm{36}×\mathrm{9}+\mathrm{36}×\mathrm{18}=\mathrm{1215} \\ $$

Commented by mr W last updated on 09/Jan/23

about the core of 2−digit numbers the core of a 2−digit number is defined as the sum of digits of the 9 times of this number. example n=25 9n=9×25=225 2+2+5=9 that means the core of 25 is 9. case 1: numbers with digit zero n=a0 9n=10n−n=a00−a0 the first digit of is a−1, the second digit is 10−a, the third is 0. sum of all digits is a−1+10−a+0=9 that means the core of a0 is always 9. case 2: numbers without digit zero n=ab 9n=10n−n=ab0−ab the third digit is 10−b the second digit is b−1−a if b−1≥a or 10+b−1−a if b−1<a the first digit is a if b−1≥a or a−1 if b−1<a sum of all digits is 10−b+b−1−a+a=9 if b−1≥a or 10−b+10+b−1−a+a−1=18 if b−1<a that means the core of ab is { ((9 if a<b)),((18 if a≥b)) :}

$$\underline{{about}\:{the}\:{core}\:{of}\:\:\mathrm{2}−{digit}\:{numbers}} \\ $$$${the}\:{core}\:{of}\:{a}\:\mathrm{2}−{digit}\:{number}\:{is} \\ $$$${defined}\:{as}\:{the}\:{sum}\:{of}\:{digits}\:{of}\:{the} \\ $$$$\mathrm{9}\:{times}\:{of}\:{this}\:{number}.\: \\ $$$${example}\:{n}=\mathrm{25} \\ $$$$\mathrm{9}{n}=\mathrm{9}×\mathrm{25}=\mathrm{225} \\ $$$$\mathrm{2}+\mathrm{2}+\mathrm{5}=\mathrm{9} \\ $$$${that}\:{means}\:{the}\:{core}\:{of}\:\mathrm{25}\:{is}\:\mathrm{9}. \\ $$$$ \\ $$$${case}\:\mathrm{1}:\:{numbers}\:{with}\:{digit}\:{zero} \\ $$$${n}=\underline{{a}\mathrm{0}} \\ $$$$\mathrm{9}{n}=\mathrm{10}{n}−{n}={a}\mathrm{00}−{a}\mathrm{0} \\ $$$${the}\:{first}\:{digit}\:{of}\:{is}\:{a}−\mathrm{1},\:{the} \\ $$$${second}\:{digit}\:{is}\:\mathrm{10}−{a},\:{the}\:{third}\:{is}\:\mathrm{0}. \\ $$$${sum}\:{of}\:{all}\:{digits}\:{is}\:{a}−\mathrm{1}+\mathrm{10}−{a}+\mathrm{0}=\mathrm{9} \\ $$$${that}\:{means}\:{the}\:{core}\:{of}\:\underline{{a}\mathrm{0}}\:{is}\:{always}\:\mathrm{9}. \\ $$$$ \\ $$$${case}\:\mathrm{2}:\:{numbers}\:{without}\:{digit}\:{zero} \\ $$$${n}=\underline{{ab}} \\ $$$$\mathrm{9}{n}=\mathrm{10}{n}−{n}={ab}\mathrm{0}−{ab} \\ $$$${the}\:{third}\:{digit}\:{is}\:\mathrm{10}−{b} \\ $$$${the}\:{second}\:{digit}\:{is}\:{b}−\mathrm{1}−{a}\:{if}\:{b}−\mathrm{1}\geqslant{a} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{or}\:\mathrm{10}+{b}−\mathrm{1}−{a}\:{if}\:{b}−\mathrm{1}<{a} \\ $$$${the}\:{first}\:{digit}\:{is}\:{a}\:{if}\:{b}−\mathrm{1}\geqslant{a} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{or}\:{a}−\mathrm{1}\:{if}\:{b}−\mathrm{1}<{a} \\ $$$${sum}\:{of}\:{all}\:{digits}\:{is}\: \\ $$$$\:\:\:\:\:\:\:\:\mathrm{10}−{b}+{b}−\mathrm{1}−{a}+{a}=\mathrm{9}\:{if}\:{b}−\mathrm{1}\geqslant{a} \\ $$$$\:\:{or}\:\mathrm{10}−{b}+\mathrm{10}+{b}−\mathrm{1}−{a}+{a}−\mathrm{1}=\mathrm{18}\:{if}\:{b}−\mathrm{1}<{a} \\ $$$${that}\:{means}\: \\ $$$${the}\:{core}\:{of}\:\underline{{ab}}\:{is}\:\begin{cases}{\mathrm{9}\:{if}\:{a}<{b}}\\{\mathrm{18}\:{if}\:{a}\geqslant{b}}\end{cases} \\ $$

Commented by mr W last updated on 09/Jan/23

$${thanks}! \\ $$

Commented by Rasheed.Sindhi last updated on 10/Jan/23

$$\mathcal{X}{cellent}\:\mathcal{S}{ir}!\:\mathcal{T}\boldsymbol{{hanks}}\:\boldsymbol{{a}}\:\boldsymbol{{lot}}. \\ $$

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