A palindrome is a number that is the same when read forwards and backwards, such as $43234$. What is the smallest five-digit palindrome that is divisible by $17$?

Guest Aug 15, 2023

#1**+1 **

I wrote some computer code, and I found that 11611 is the smallest palindrome divisible by 17.

In case you are curious, here is the code I wrote for this particular problem. Note I did not finish the code because I already found the smallest palindrome with the incomplete code.

int startNum = 10001;

int numToTest = startNum;

for (int i = 0; i < 10; ++i){

for (int j = 0; j < 10; ++j){

if (numToTest % 17 == 0){

System.out.println(numToTest + " is divisible by 17");

}

else{

System.out.println(numToTest + " is not divisible by 17");

}

numToTest = numToTest + 100;

}

startNum = startNum + 1010;

numToTest = startNum;

}

The3Mathketeers Aug 16, 2023

#1**+1 **

Best Answer

I wrote some computer code, and I found that 11611 is the smallest palindrome divisible by 17.

In case you are curious, here is the code I wrote for this particular problem. Note I did not finish the code because I already found the smallest palindrome with the incomplete code.

int startNum = 10001;

int numToTest = startNum;

for (int i = 0; i < 10; ++i){

for (int j = 0; j < 10; ++j){

if (numToTest % 17 == 0){

System.out.println(numToTest + " is divisible by 17");

}

else{

System.out.println(numToTest + " is not divisible by 17");

}

numToTest = numToTest + 100;

}

startNum = startNum + 1010;

numToTest = startNum;

}

The3Mathketeers Aug 16, 2023

#2**0 **

To find the smallest five-digit palindrome that is divisible by 17, we can start by considering the multiples of 17 until we find a palindrome.

First, we need to find the smallest multiple of 17 that is a five-digit number. To do this, we can divide 10000 (the smallest five-digit number) by 17:

\[10000 \div 17 \approx 588.23.\]

So, the smallest multiple of 17 that is a five-digit number is \(17 \times 588 = 9996\).

Next, let's check the next multiples of 17 to find a palindrome:

\[17 \times 589 = 9993\]

\[17 \times 590 = 10030\]

\[17 \times 591 = 10047\]

The next multiple, \(17 \times 592\), is a five-digit number and is equal to \(99944\). This is a palindrome, so the smallest five-digit palindrome that is divisible by 17 is \(99944\).

SpectraSynth Aug 16, 2023