A number is called a palindrome if it reads the same backward as well as forward. For example 285582 is a six digit palindrome. The number of six digit palindromes, which are divisible by 55, is ____________.
Answer (integer)
100
Solution
<table class="tg">
<thead>
<tr>
<td class="tg-baqh">5</td>
<td class="tg-baqh">a</td>
<td class="tg-baqh">b</td>
<td class="tg-baqh">b</td>
<td class="tg-baqh">a</td>
<td class="tg-baqh">5</td>
</tr>
</thead>
</table>
<br><br>For divisible by 55 it shall be divisible by 11 and 5
both, for divisibility by 5 unit digit shall be 0 or 5 but
as the number is six digit palindrome unit digit is 5.
<br><br>A number is divisible by 11 if the difference between sum of the digits in the odd places and the sum of the digits in the even places is a multiple of 11 or zero.
<br><br>Sum of the digits in the even place = a + b + 5
<br><br>Sum of the digits in the odd places = a + b + 5
<br><br>Difference between the two sums = (a + b + 5 ) - (a + b + 5) = 0
<br><br>0 is divisible by 11.
<br><br>Hence, 5abba5 is divisible by 11.
<br><br>So, required number = 10 $\times$ 10 = 100
About this question
Subject: Mathematics · Chapter: Permutations and Combinations · Topic: Fundamental Counting Principle
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