it is a five digit whole number.it is an even palindrome.it is less than 30000.it is divisible by 4 and 9?

Question

it is a five digit whole number.it is an even palindrome.it is less than 30000.it is divisible by 4 and 9, but none of its digits are divisible by three.

Answer 1

I got it!
A 5 digit palindrome has digit pattern abcba.
Conditions of the problem dictate that
10b + a must be divisible by 4, i.e.,
a + 2b must be divisible by 4,
and
2a + 2b + c must be divisible by 9.
Further, none of a,b,c is divisible by 3.
This tells us that a is even and since the number
is less than 30000, a must be 2.
The possibilities for b are 1, 5 or 7.
Trying b = 1 forces c to be divisible by 3.
But b = 5 works:
4 + 10 + c is divisible by 9, so we can take c = 4.
The answer is 25452.
Unfortunately, the other answer given, 27072, is
invalid because 0 is divisible by 3.

Answer 2

Let's break this down and see if we can figure it out....

"5-digit whole number" that's a palindrome (reads same backwards as forwards) can be written as this:

10,000A + 1,000B + 100C + 10B + A

Knowing it's less than 30,000 tells us that A is either 1 or 2

So our number is either 1BCB1 or 2BCB2

If it's divisible by 4, then the number has to be even, so we know our number is 2BCB2 (since 1BCB1 is odd).

Now we just need to solve for B and C!

We should use divisibility by 9 as a clue, as well as the fact that neither B nor C is divisible by 3.

If the number is divisible by both 4 and 9, then that also means it's divisible by 36 (4 x 9).

At this point, I would just "brute force" it and look at all the possible combinations (there's not that many) and see which ones are divisible by 36. Substitute all the digits from 0-8 except 3 and 6 for B and C, like thus:

20002
20102
20202
20402
20502
20702
20802

21012
21112
21212
21412
21512
21712
21812

etc. (Ugh, that's ugly isn't it? Let me work on this some more!)

OK, I just plugged all the possibilities into an excel spreadsheet, and the only two possible answers are 25,452 and 27,072, as the next two people below me have already determined.

I hope my explanation of how to tackle this problem helped.

Answer 3

Palindrome For Even

Answer 4

27072

heres the list

http://mathforum.org/library/drmath/view/57170.html

Answer 5

27072 seems to fit.

Answer 6

25452?