Am I correct?

Apr 2009
288
56
Got a challenging question (for me) that I want to ensure that I am correct.

Question
The numbers 828 and 313 are 3-digit palindromes where 828-313=515 which is also a palindrome. How many pairs (a, b) of 3-digit palindromes are there with a>b and with a-b also a 3-digit palindrome?

My Answer: 1980
Reasoning
Conditions
1. For palindromes aba and cdc to for a palindrome, then b>d.
2. To form a 3-digit palindrome, then a>1.

So we check a case by case analysis, starting with a=9 and varying b \(\displaystyle (0\leq{b}\leq{9})\)
Number: 909 919 929 939 .... 999
Pairs: 8 16 24 32 .... 80

Moving onto a=8
Number: 808 818 828 ..... 898
Pairs: 7 14 21 ..... 70

And so on to a=2
Number: 202 212 .... 292
Pairs: 1 2 .... 10

So we a pattern of
8x(1+2+3+...+10) + 7x(1+2+3+...+10) + 1x(1+2+3+....10)
This becomes
(1+2+3+...+10)(1+2+3+...+8)=55x36
Ans=1980

So could someone please check this.
Thanks
 
Jul 2019
1
0
Australia
Yes, you are correct. it is 1980.
=(1+2+3+4+5+6+7+8+9+10)x(1+2++3+4+5+6+7+8)
=55x36
=1980

And I think it is from the 2008 AMC exam paper.