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# palindromes

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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 $17$?

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;
}

Aug 16, 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
#2
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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 $$99944$$.

Aug 16, 2023