Palindrome Fair

Number Theory Level 2

A palindromic number is a number that reads the same both forwards and backwards. For example, 54045 is a 5-digit palindromic number.

There is a square 3-digit palindromic number, which is also palindromic when divided by 2. What is the number?

The answer is 484.

2 solutions

Adarsh Kumar
Jun 20, 2014

Because the no. is a palindrome it must be of the form 'aba' but there is one more condition that needs to be satisfied and that is it should be a perfect square there is one more condition and that is when the no. is divided by 2 it should be a palindrome.The first number that satisfies the first two conditions is 121 but that is not our no. because it is not divisible by 2.The second number is 484 which satisfies all the conditions.

A little different approach. The number should be divisible by 4 (square of even number) and in ABA form. (Tip: squares of even numbers should have either 4 or 6 as unit digit). Only two such numbers are 484 and 676; the latter doesn't meet the special condition. Alternatively, the solution can be manually obtained by squaring the numbers between 10 and 31 (squares are 100 and 961 respectively).

Chrishan Silva - 6 years, 11 months ago

Victor Paes Plinio
Sep 26, 2014

The number must be $100a+10b+a$ .

The number divided by 2 is $50a+5b+0.5b$

This is a perfect square palindrome. We know that: The square of a palindrome is an another palindrome. Have only nine 2-digt palindromes $11, 22, 33, 44, 55, 66, 77, 88 and 99$ .

We need only the smallest one. The number is $11$ and ${11}^{2}=121$ . $121$ is odd. Dividing by 2 we get $60.5$ , not a palindrome.

Now we know that: we need the smallest 2-digit even palindrome, or $22$ .

${22}^{2}=484$ . $484$ is a palindrome and $484/2=242$ is a palindrome too.

So the answer is $\boxed {484}$

This is true: "The square of a palindrome is an another palindrome." But the converse is not.

Dan Lawson - 6 years, 8 months ago

Palindrome Fair

The answer is 484.

2 solutions

0 pending reports