Strings: Level 3 Challenges


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 \(123+12+1 = 136\) 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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