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Problem 10) (3 points): Prove that any set of 10 distinct positive inte gers contains a pair whose difference is divisible by 9....

Question

Problem 10) (3 points): Prove that any set of 10 distinct positive inte gers contains a pair whose difference is divisible by 9.

Problem 10) (3 points): Prove that any set of 10 distinct positive inte gers contains a pair whose difference is divisible by 9.



Answers

Prove that each statement is true for all positive integers.
$9^{n}-1$ is divisible by 8

A number is divisible by nine. If the sum of its digits is divisible by nine, for example, we have 45 4 plus five is nine. That means this is divisible by nine. Same with 72 7 plus two is nine and nine is divisible by nine. Then we have a few that are not. For example, one plus five plus eight is 14 and 14 is not divisible by nine. However, if you have three and seven and eight, you get 18 and 18 is divisible by nine. The only other weather other one on this list that is divisible by nine is 585. Every other number has a total that is not divisible by nine. So that's end of it. Only four numbers.

We need to prove that 10 power and plus three into four power and plus two plus five is actually divisible by nine for all and belongs to national numbers. So we need to put that it is divisible by nine. Let's use mathematical induction. So assume that this is a statement be off. And so what is the statement the often 10 power and plus three into £4 plus two plus five is divisible by nine. So for mathematical induction we need to Test the bass case be of one. So be of one is 10 power one. That's 3 to 4 power one plus 2 34. Part three plus five. So that will be 10 plus three into 64 which is 1 92 plus five. So it is 1 92 plus 15. 1 92 plus 15 will be how much? +207 207 is divisible by nine because two plus zero plus seven sum of the digits is divisible by nine. The number is divisible by nine. So yes you're funny strips the base case is true. Now what we need to do assumption case. So let me off baby too. We're assuming that the off case. True. So that means we are resuming 10 parquet. Let's three into four. Park a place to Plus five is a multiple of nine. I'll call it has some 90 for some interior cute. So now let's Verify whether Beyonce gave us one Is also a multiple of nine. Using the previous assumption Kiss. So what is your keep this on its stand park a plus one plus three into four parquet plus one plus two. We just Give me three plus 5 so that I can write it as. So 10 power K. Plus one Plus 13-4. Power cable is three plus. Fight this fight I can replace with from this equation I can write fires nine Q minus 10 parquet minus stream to four. Park a place to. Alright I replaced this five with this. It will be nine Q. Uh huh minus 10 baki -3 times of four power cable is too. So I can take 10 park a common in these two cities, 10 parquet into 10 -1. And I think I can take four power K plus two common In this and this. So when I take four park a place to common, I'll be getting 4 -1. And this is nine cube. So basically I'm beginning nine into 10 party Plus three in 2019- four. Park a place to plus nine into Q. And you can see that nine is common everywhere. So it is 19 to 10 parquet. It's for parquet plus two plus cube which is a multiple of nine. Yes. So keep this one is true, so since B of A is true, that implies B. Of this one is true. So for all, and belongs to end your fantasies by principle of mathematical induction.

When n equals one, the statement says that night is a factor of 135, which is true next resume. The statement is true for an eccles cake. Then for N equals K plus one it becomes 10 to the power of K-plus two Past three times turned to a power of K plus one plus five which he goes 10 times 10 to the power of K plus one Plus 10 times three times 10 to the power of cake Plus 10 times five. My nurse 44 And there's he calls 10 times. Change your powerful K plus one plus three times 10 to the power of K Plus 5 -45. Note that the statement is true for any calls case and has the first term. Night is a factor of the first term term and also night is a factor of 45. Hence neither factor of this expression, which means the statement is true for ecos K plus one hands. By mathematical induction. The statement is true for every end, which is a positive integer.

When n equals one, the statement says that night is a factor of 135, which is true next resume. The statement is true for an eccles cake. Then for N equals K plus one it becomes 10 to the power of K-plus two Past three times turned to a power of K plus one plus five which he goes 10 times 10 to the power of K plus one Plus 10 times three times 10 to the power of cake Plus 10 times five. My nurse 44 And there's he calls 10 times. Change your powerful K plus one plus three times 10 to the power of K Plus 5 -45. Note that the statement is true for any calls case and has the first term. Night is a factor of the first term term and also night is a factor of 45. Hence neither factor of this expression, which means the statement is true for ecos K plus one hands. By mathematical induction. The statement is true for every end, which is a positive integer.


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