Question Number 83910 by redmiiuser last updated on 07/Mar/20
$$\mathrm{find}\:\mathrm{all}\:\mathrm{6}\:\mathrm{digit}\:\mathrm{numbers}\:\mathrm{which}\:\mathrm{are}\:\mathrm{not} \\ $$$$\mathrm{only}\:\mathrm{palindrome}\:\mathrm{but}\:\mathrm{also}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{495}. \\ $$
Commented by redmiiuser last updated on 08/Mar/20
$$\mathrm{CAN}\:\mathrm{ANYONE}\: \\ $$$$\mathrm{ANSWER}\:\mathrm{THIS} \\ $$$$\mathrm{QUESTION}. \\ $$
Answered by Kunal12588 last updated on 08/Mar/20
$${there}\:{are}\:\mathrm{11}\:{such}\:{numbers} \\ $$$$\mathrm{504405},\:\mathrm{513315},\:\mathrm{522225},\:\mathrm{531135},\:\mathrm{540045}, \\ $$$$\mathrm{549945},\:\mathrm{558855},\:\mathrm{567765},\:\mathrm{576675},\:\mathrm{585585}, \\ $$$$\mathrm{594495} \\ $$
Commented by redmiiuser last updated on 08/Mar/20
$${GREAT} \\ $$
Commented by naka3546 last updated on 08/Mar/20
$${Show}\:\:{your}\:\:{workings}\:,\:\:{please}\:. \\ $$
Commented by Kunal12588 last updated on 08/Mar/20
$${i}\:{don}'{t}\:{think}\:{i}\:{can}\:{find}\:{it}\:{with}\:{any}\:{method}. \\ $$$${I}\:{just}\:{used}\:{the}\:{bruteforce}\:{method}. \\ $$
Answered by redmiiuser last updated on 08/Mar/20
$${there}\:{is}\:{an}\:{easy}\:{way} \\ $$$${we}\:{know}\: \\ $$$$\mathrm{495}=\mathrm{9}×\mathrm{5}×\mathrm{11} \\ $$$${as}\:{the}\:{number}\:{must} \\ $$$${be}\:{divisible}\:{by}\:\mathrm{495}\:{hence} \\ $$$${so}\:{by}\:\mathrm{9}\:{and}\:\mathrm{5}\:{and}\:\mathrm{11}. \\ $$$${to}\:{be}\:{divisible}\:{by}\:\mathrm{5}\: \\ $$$${the}\:{last}\:{dight}\:{must}\:{be} \\ $$$$\mathrm{5}\:{and}\:{not}\:\mathrm{0}. \\ $$$${again}\:{for}\:\mathrm{9}\:{the}\:{sum}\:{of} \\ $$$${the}\:{dights}\:{must}\:{be}\:{divisi}\mathrm{6}{le} \\ $$$${by}\:\mathrm{9}. \\ $$$${now}\:{for}\:\mathrm{11}\:{lets}\:{take}\: \\ $$$${an}\:{example}. \\ $$$$\mathrm{594495}\: \\ $$$${here}\:{the}\:{number}\:{is} \\ $$$${divisible}\:{by}\:\mathrm{5}\:{and}\:\mathrm{9}. \\ $$$${but}\:{the}\:{number}\:{is}\:{divisible} \\ $$$${by}\:\mathrm{11}\:{on}\:{applying}\:{the} \\ $$$${divisiblity}\:{trick}\:{i}\:{get}\:{this}. \\ $$$${again}\:{in}\:\mathrm{495594} \\ $$$${the}\:{number}\:{is}\:{divisible}\:{by} \\ $$$$\mathrm{11}{and}\mathrm{5}{and}\mathrm{9}. \\ $$$${but}\:{the}\:{change}\:{is}\: \\ $$$${the}\:{terminal}\:{numbers} \\ $$$${are}\:{decreased}\:{by}\:\mathrm{1}\: \\ $$$${while}\:{middle}\:{ones}\: \\ $$$${are}\:{increased}\:{by}\:\mathrm{1}. \\ $$$${hence}\:{by}\:{applying} \\ $$$${this}\:{concept}\:{we}\:{can}\:{solve} \\ $$
Commented by redmiiuser last updated on 08/Mar/20
$${i}\:{am}\:{sorry}\:{it}\:{must}\:{be} \\ $$$$\mathrm{549945}\:{not}\:\mathrm{495594} \\ $$