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 ____________.

Solution

Maths sir · 4 steps

Write the palindrome as abccba: divisibility by 11 comes free, and 5 fixes the first digit.

1. Set up the form. A six-digit palindrome looks like abccba̅. The last three digits mirror the first three, so it is fixed once you choose a, b and c, with a ≠ 0.
2. Test for 11. Take the alternating sum of digits from the left: a - b + c - c + b - a = 0. Zero is a multiple of 11, so every six-digit palindrome is divisible by 11.
3. Test for 5. The last digit a must be 0 or 5. But a is also the first digit, so it cannot be 0. That leaves a = 5 only.

   1ₐ₌₅ × 10_b × 10_c = 100

4. So. b and c can each be any digit from 0 to 9, so there are 100 such palindromes.

The common slip: 200. Allowing a = 0 as well as a = 5 gives 200. With a = 0 the number starts with 0 and is not six-digit at all.

The idea to keep. Any even-length palindrome is a multiple of 11, because its alternating digit sum is always zero. In a palindrome the last digit is also the first, so it can never be 0.

Answer from the NTA final answer key. Solution by PrepWiser.