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How to prove a no is divisible by 9...

Question

How to prove a no is divisible by 9

How to prove a no is divisible by 9



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

So we have to show that nine power and plus 1 -8 and -9 is divisible by 64 whenever and is a positive integer. So let's see the formula for the binomial expansion of one plus X. Power in S and C. Note plus. And even into X plus answer tuned to X square tell N C. N. Expert. And so let's consider nine power and plus one which can be written as one plus eight power and plus one. Not using the above formula. We can expand 1% power and plus one has and plus once a nod Plus and plus 178 Plus and plus wants you to 80 square plus and plus one C. +38 cube still and plus one C. N. +18 and plus one. Okay soon. Nine power and plus one can written as no value of N plus one C. Notice one The value of endless 1 C1 is and plus one and do it plus And plus one. So you too 80 square And plus one c. 3 a cube till 10 plus one C. N. +18 and power one nine power and plus one equals 2. one Plus eight. n plus eight plus. No let's take out 80 square comin from a return now it will become a T square into end Plus one say to Plus and plus one c. 3 in 28 Still and plus one c. and plus one if you'd bothered. And my husband now let's bring eight and this one and this it will become nine. So nine power and plus one -8 10- and will become 64 into some number K. And this game will be a positive number and a natural number. We are K. as equals two and plus once here too And plus one c. 38 till and plus one C. And +18 Power and machine. So we can clearly see That nine power and plus 1 -8 10 minus nine, Is a number which is divisible by 64. I hope you understand the solution. Thank you.

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.

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.

So this question we have that if a isn't inter, just such a three is not a divisor. Then three is a device off a +18 plus two. So if three is not a device called a what this means is that we can write what we can't write. A is not three times by some emergency where season. Instead, it means I can't do this. But what it does mean, though, is it means that we write a as being three c plus one or greasy plus by two for see an integer. Now we can't do three C plus by three, because we can look under this because we can factor out three here C plus one, and this imply that three is a survivor, so we're only left with the suitcases. So in case one, if we substitute this onto this side, we have greasy plus by one plus one plus three C plus my one plus much, this thing becomes three seats plus by two these things. This thing becomes three C plus by three. Then you can talk her out of your contract out a three D. You get three C plus way too, as I see plus one. So you know, this is an interview. You know, this is introduction. So this whole thing isn't it Up and three, you can clearly see the fact that so three divides this whole thing. Not for this case we do the same thing. So we have three seat plus y two plus my one kind of three C plus by two plus two we seen becomes greasy despite three misses. Three c plus by four. We can factor out a three. She's really see if that's what one greasy fast before. So once again this thing is it's just we haven't well, we have three times by Internet. So therefore, three is a factor of this. Enhances of backing off this. So the results prove


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