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## Strings

Strings are basically "words in computers". As an ordered set of characters, these are the building blocks that allow us to do things from searching filesystems to decrypting ciphers.

# Level 3

Let the reverse of a positive integer $$n$$, denoted $$R(n),$$ be the result when the digits of the number are written backwards; for example, $$R(190) = 091,$$ or just $$91.$$

Call a positive integer $$n$$ brilliant if $n + R(n)$

is a multiple of 13. Let $$B$$ be the $$10000$$th brilliant number. Compute the last three digits of $$B.$$

Given a natural number, we start $$\color{Blue}{\text{eating the number}}$$ from either left or right i.e. we start removing its digits one by one from left to right, or from right to left.

We define a set $$\color{Blue}{\text{dish}}$$ of the number, which is obtained by noting the number formed after eating every digit (The original number is also included) .

The $$\color{Blue}{\text{taste}}$$ of a number is sum of all numbers in that number's dish.

What is the smallest non-palindromic number which when eaten from left gives same taste as eating from right?

Details and Assumptions:

• The dish of a number can be obtained in 2 ways, either eating from left or eating from right and hence there'll be 2 tastes for each number (maybe the same, that's where you count the number!)

• Example of a dish, dish of the number 12635 as eaten from left will be $$\{12635,2635,635,35,5\}$$ and its dish when eaten from right will be $$\{12635,1263,126,12,1\}$$

• Taste of the number 123 will be $$123+23+3 = 149$$ from left and it will be $$123+12+1 = 146$$ from right.

• A non-palindromic number is the one which is not the same when read from left or from right, e.g. $$12321 , 22 , 1441, 8$$ are some examples of palindromic numbers, whereas $$98,234,239478$$ are some non-palindromic numbers.

A palindromic-prime or PalPrime is a prime number that is also a palindrome.The first few PalPrimes are $$2, 3, 5, 7, 11, 101, 131, 151, 181, 191...$$. Let $$S$$ be the sum of the digits of the largest PalPrime $$N$$ such that $$N<10^9$$.What is the value of $$S$$?

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