How many 3 digit palindromes are multiples of 3?

Answer 1

0

Answer 2

please list ALL three-digit palindromes that are multiples of three. Show your work and justify your answers.

111

141

171

222

252

282

303

333

363

393

414

444

474

525

555

585

606

636

666

696

717

747

777

828

858

888

909

939

969

999

all of these are because each digit in each of these number adds up to a multiple of 3 (like 3,6,9,etc.)

Answer 3

A three-digit palindrome is a number of the form aba.

Lets assume you will not accept 000, 030, 060, and 090 as valid.

Then the number is divisible by three if the sum of the digits is divisible by three.

So for each choice of a (there are 9 possible) we must solve 2a + b = some multiple of 3. For the more mathematically inclined, 2a + b = 0(modulo 3).

Now for the difficult bit - There are always exactly 3 answer to this equation! I'll illustrate by example:
Suppose we take a = 5, 2a = 10, we want b between 0 and 9 so that 10 + b is a multiple of 3; 10 + b will of course be between 10 and 19; the multiples of 3 are 12, 15, 18, giving b as 2, 5, or 8, and our palindromes as 525 555 585.

Its not too hard to see that in any range of 10 numbers there are always exactly 3 multiples of three.

This gives us 27 palindromic numbers (as I said, not counting 000, 030, 060 and 090)