Number-theory-A-palindrome-is-a-number-that-reads-the-same-backwards-as-forwards-as-3141413-a-How-many-two-digit-palindromes-are-there-b-How-many-three-digit-ones-c-How-many-k-digits-ones-

Question Number 134960 by bobhans last updated on 09/Mar/21

$$\mathrm{Number}\:\mathrm{theory} \\ $$A palindrome is a number that reads the same backwards as forwards, as 3141413.
(a)How many two-digit palindromes are there?
(b)How many three-digit ones?
(c)How many k-digits ones?

Answered by mr W last updated on 09/Mar/21

$$\left({c}\right)\:{k}−{digit}\:{numbers} \\ $$$${case}\:\mathrm{1}:\:{k}=\mathrm{2}{n} \\ $$$$\mathrm{XYYY}…\mathrm{YY}…\mathrm{YYYX} \\ $$$$\mathrm{X}:\:\mathrm{1}−\mathrm{9} \\ $$$$\mathrm{Y}:\:\mathrm{0}−\mathrm{9} \\ $$$$\Rightarrow{number}\:{of}\:{numbers}=\mathrm{9}×\mathrm{10}^{{n}−\mathrm{1}} \\ $$$${case}\:\mathrm{2}:\:{k}=\mathrm{2}{n}+\mathrm{1} \\ $$$$\mathrm{XYYY}…\mathrm{YZY}…\mathrm{YYYX} \\ $$$$\mathrm{X}:\:\mathrm{1}−\mathrm{9} \\ $$$$\mathrm{Y}:\:\mathrm{0}−\mathrm{9} \\ $$$$\mathrm{Z}:\:\mathrm{0}−\mathrm{9} \\ $$$$\Rightarrow{number}\:{of}\:{numbers}=\mathrm{9}×\mathrm{10}^{{n}} \\ $$$${generally}\:\mathrm{9}×\mathrm{10}^{\lfloor\frac{{k}+\mathrm{1}}{\mathrm{2}}\rfloor−\mathrm{1}} \\ $$$${k}=\mathrm{2}:\:\mathrm{9}×\mathrm{10}^{\mathrm{0}} =\mathrm{9}\:{numbers} \\ $$$${k}=\mathrm{3}:\:\mathrm{9}×\mathrm{10}^{\mathrm{1}} =\mathrm{90}\:{numbers} \\ $$

Commented by Rasheed.Sindhi last updated on 09/Mar/21

$$\mathcal{W}{onderful}\:\:\boldsymbol{{sir}}! \\ $$

Commented by mr W last updated on 09/Mar/21

$${thanks}\:{sir}! \\ $$

Commented by mr W last updated on 09/Mar/21

$${it}\:{seems}\:{you}\:{are}\:{not}\:{so}\:{often}\:{here}\:{as} \\ $$$${before}. \\ $$

Commented by Rasheed.Sindhi last updated on 09/Mar/21

$${Yes}\:\boldsymbol{{sir}}\:{you}'{re}\:{right}.\:{I}'{m}\:{not}\:{able} \\ $$$${to}\:{concentrate}\:{much}. \\ $$

Commented by bobhans last updated on 09/Mar/21

$$\mathrm{yess} \\ $$

Commented by liberty last updated on 09/Mar/21

$$\mathrm{how}\:\mathrm{many}\:\mathrm{four}\:\mathrm{digit}\:\mathrm{ones}\:\mathrm{sir}? \\ $$$$\mathrm{k}=\mathrm{2}×\mathrm{2}\:\Rightarrow\mathrm{n}=\mathrm{2} \\ $$$$\mathrm{the}\:\mathrm{number}\:=\:\mathrm{9}×\mathrm{10}^{\mathrm{2}−\mathrm{1}} \:=\:\mathrm{90}? \\ $$

Commented by mr W last updated on 10/Mar/21

$${yes} \\ $$

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