The number $300$ is palindromic in bases 7,8 and 9. $300_{10} = 606_{7} = 454_{8} = 363_{9}$ .

What is the total number of bases in which the representation of $300_{10}$ would be palindromic with more than one digit.

9
18
16
3
15
4
13

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$300$ being divisible by $2,3,4,5,6$ cannot be a palindrome in any of those bases.

It is already given that it is a palindrome in bases $7,8,9$ .

Searching for 3 digit palindromes of type $\overline{xyx}_b$ , with base $b \le \lfloor\sqrt{300} \rfloor = 17$ , We get the equation as $(b^2+1)x+by=300$ . $0 \le x,y < b$ . Solving,

we find valid solutions for base 13. $300_{10}=1a1_{13}$

Now with 2 digit palindromes, i..e $18 \leq b < 300$ , the required equation becomes

$(b+1)x=300$ , which is satisfied for $b=19,24,29,49,59,74,99,149,299$

Thus, there are $\boxed{13}$ such bases.