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 minimum value of $N$ is $\qquad$ (answer in decimal).
Number Theory, Bases and Place Values.
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, $N=a \times 14+a=b \times 20+b$;
$$
N=15 a=21 b ;
$$

Then $N$ is a multiple of both 15 and 21;
Thus, $N$ is a multiple of $[15,21]=3 \times 5 \times 7=105$;

And $(105)_{10}=(77)_{14}=(55)_{20}$, which meets the requirements;
Therefore, the minimum value of $N$ is 105.