Here is one that I dont understand how to do.?

Question

Use the Pigeonhole Principle to explain why any collection of 12 positive integers must contain two numbers whos difference is divisible by 10. Your answer should make it clear that you understand what the Pigeonhole Principle says (you may express it informally).

If got a general idea, that since there are 12 numbers, there must be two numbers that add up to form a multiple of ten because there are only 10 single unit digits, and there only needs to be 2 other digits to make a multiple of ten. Example: Take the numbers 1,3,6,10,12,13,18,20,21,22,25, and 27. Here, since 27 and 3 add up to 30, and 30 is divisible by 10... but that wont always work, such as if you take 11,21,31,41,51,61,71,81,91,101,111, and 121.

Help?

Whoa, my bad.

Answer 1

To have a difference of 10, the unit digit must be the same. So to rephrase the question, is it possible to list 12 positive integers, each with a different unit digit. Keeping in mind the potential choices are 0,1,2,3,4...,9. So there is no list of 12 numbers such that every number has a different digit in the "ones" position. Therefore there is a potential answer.

The pigeon hole principle is used because we have 10 potential candidates for the "ones" position of an integer (the pigeon holes), and we have 12 numbers total (the pigeons) and because 12 > 10, there must be at least two of the 12 numbers must have a difference of 10 (at least two pigeons must share a hole).

Answer 2

Suppose you have 12 positive integers. The numbers have to end in 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9.
Since there are only 10 possible digits that the numbers can end in, that means there are at least two that end with the same number. This means there will be a pair of numbers whose difference is a multiple of 10.

Answer 3

Difference is subtraction, not addition.

For example (your list of numbers 11, 21, 31, 41, 51, 61, 71, 81, 91, 101, 121) ==> 21-11=10, 10/10=1 or 41-21=20, 20/10=2

Answer 4

Your taking the difference, NOT adding the numbers......so in your list, 21 - 11 the difference is 10, thus 10/10 = 1.

Answer 5

If I understand your question, you only need eleven numbers to ensure you have two whose difference is divisible by 10. Basically, you need two numbers that end in the same digit, and with eleven numbers, you are sure to have a pair of numbers like that.

Answer 6

I was under the impression that "difference" meant subtract the two numbers from each other. Not add.
Try it again

Answer 7

Watch out: the problem says *whose difference*, and you are adding them up. You should subtract them. That may explain the difficulties you are facing...