【Question 3】
A natural number $N$ greater than 20 can be represented as a palindrome in both base 14 and base 20 (a palindrome is a number that reads the same forward and backward, for example, 12321 and 3443 are palindromes, while 12331 is not a palindrome). The smallest value of $N$ is $\qquad$ (answer in decimal notation).
【Analysis and Solution】
Number Theory, Base and Place Value.
Since $N>20$;
Therefore, $N$ is not a single digit in base 14 and base 20;
We want $N$ to be as small as possible, so let $N=(\overline{a a})_{14}=(\overline{b b})_{20}$;
That is, $N=a \times 14+a=b \times 20+b$;
$N=15 a=21 b$;
Thus, $N$ is a multiple of both 15 and 21;
Therefore, $N$ is a multiple of $[15,21]=3 \times 5 \times 7=105$;
And $(105)_{10}=(77)_{14}=(55)_{20}$, which meets the requirements;
Hence, the minimum value of $N$ is 105.