A number like 53435 is called a palindrome. because it reads the same backwards and forwards. A friend of mine claims that all palindromes with four digits are exactly divisible by 11. Are they? How so?
2006-09-30 23:01:02 · 7 answers · asked by Anonymous in Education & Reference ➔ Homework Help
Yep, I think they are, and I'm at a loss as to why. I do, however, know that if you add all the digits of ANY number together, and the result is divisible by 3, then the original number was divisible by 3. The same goes with 9...
2006-10-01 00:27:44 · answer #1 · answered by 42ITUS™ 7 · 2⤊ 0⤋
Recall the divisibility rule for 11.
A whole number is divisible by 11 if
the sum of the digits in the odd powered places
minus the sum of the digits in the even powered
places is a multiple of 11.
This is sometimes called "the every other numbers rule".
A simpler way of looking at it is as follows:
Start at the first digit. Find the sum of every
other number. Now go to the second digit.
Find the sum of every other number.
Subtract the two sums. If the difference
is a multiple of 11, so is the original number.
Example: Is 14641 a multiple of 11?
The first sum is 8 and the second sum is also 8.
The difference is 0, which is 11*0, so the answer
is "yes".
Any 4 digit palindrome must look like a b b a.
Here the first sum is a + b and the second sum is the same.
Again, the difference is 0, which is a multiple of 11.
So your statement is true.
2006-10-01 11:51:18 · answer #2 · answered by steiner1745 7 · 0⤊ 0⤋
I actually just tried it and it worked! I think its coz with any palindrome with 4 digits has to have 2 pairs of the same number. 1 pair of the same number is always divisible by 11 eg. 33 divid by 11 is 3. 2 pairs is also divisble by 11 no matter what order their put in eg. 9559/11=a whole number. the two pairs of the same number are 9,9 and 5,5. you can do this with any number, eg. 36 is divisible by 6 and so is 12. You put these 2 numbers together and mix them up and u get 3126 which is divisible by 6.
2006-10-01 06:19:33 · answer #3 · answered by May contain traces of nuts 4 · 0⤊ 0⤋
It is also interesting to note that those palindromes with digits that increase from the left and then decrease appropriately as we get to the middle digit(s) have palindromic quotients when they are divided by 11!
2006-10-01 06:13:33 · answer #4 · answered by iyiogrenci 6 · 0⤊ 0⤋
Just wondering but I thought this term was just for words that were spelled forwards and backwards coming out the same.
2006-10-01 06:06:20 · answer #5 · answered by Mr. PDQ 4 · 0⤊ 1⤋
yes
2006-10-01 06:10:02 · answer #6 · answered by spark_walk 2 · 1⤊ 0⤋
try it yourself and tell me.....
2006-10-01 06:02:37 · answer #7 · answered by Khairudin Bin Salim 4 · 0⤊ 1⤋