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Pts) How many positive integers ess than 100Olare divisible by 5lor 17hHow many positive integers ess than 10Ol are not divisible by either slor 17?How many positiv...

Question

Pts) How many positive integers ess than 100Olare divisible by 5lor 17hHow many positive integers ess than 10Ol are not divisible by either slor 17?How many positive integers ess than 1ood are divisible by 17but not by 5h

pts) How many positive integers ess than 100Olare divisible by 5lor 17h How many positive integers ess than 10Ol are not divisible by either slor 17? How many positive integers ess than 1ood are divisible by 17but not by 5h



Answers

Find the number of positive integers not exceeding 100 that are not divisible by 5 or by 7 .

So the number of plastic number is pitiable off. 3 17. 95. Noticed that the energy that are multiples off 30 city off. 3. 17. 51. We see the created 705 GCT off 17 30. Published president by However, the three d. C. D off 17. 35 is exceed 1000 at least 1 7085. So thes three D c. D guest in 1000, because it will self destruct them. So we have ah, Horan's off three in 17 1035. It's about 383 8 28 respectively. And then you have to simplify it. So 33% plus we get management management management that is equals stranded 90. So the number of positive integers less than 1000. And that leaves visible but 3. 17 or 35 This 1000 minus 3 98 Sequels toe 610 in three years

Seven and leaves that. How many? Three digit numbers that are divisible by five can be formed using video, 0235 and seven. If nor did it can be used one modern months. So the thing that's the important thing for us to discuss is that how to know when the number is divisible by five. So the device ability test for number two divisible by five. Simple. That the numbers should end with a zero or it should end with a five mile number ends with zero or five. The number is automatically automatically, they divisible by five. Here, there's one more thing to be very to worry about that. You know, zero isn't the number And it says that use numbers should be a three digit number. You should be mindful of the fact that you're three digit numbers should not start with zero because if it doesn't start with 00 in the start of the number has no significant significance. Therefore the three digit number will simply simply I could get number if it is starting with zero with this clear, let's see, I have Yeah. Yeah, would have been there 1, 2 and three and fourth. And these are two kinds of possibilities. Uh And this is like I'm made to this is a way to visualize just keep it. So let's say your number is ending with zero for the first case and ending with the five for the second case, which simply means mhm Offly means that you you have one when possibilities in worse scenarios for your last day for your life. But when your number is starting with Ending with zero, you will have 1234. You will have four possibilities When your number is starting its ending with zero And three person ready for the second agent. But the case when we see many number is starting with five, ending with five because when it is ending with five, you cannot have zero as your first digit here, which means that you will have only three possibilities for your first digit in this scenario and again three K. And now you will have again three possibilities for the second digit for the second scenario. Because now you can have zero included. So now if you calculated four into three is simply 12 and three into 3 is nine. And if if you add these votes scenarios, you will have 21, Which is option three of the possible answers. So I hope the explanation was helpful and clear. Thank you for your time. Mhm. Yeah.

Alright for part a We want to answer the question how many positive integers between 10,909 99 Excuse me, are inclusive? Are divisible by nine. So what we can do here is figure out first how many positive integers? Less than 9999. I suppose less than or equal to Are divisible by nine. So we do the floor of 9,999 divided by nine. It's going to be an integer anyways, but that is 1,111. Then we need to subtract off the number of integers Less than 1000 that are divisible by nine. Yeah. And that Is going to be 1 2nd here. 111. So The answer is going to be 1111 -111 which is of course equal to 1000. Uh For the next part we want to answer how many are even so to find to find the number of even numbers, even positive integers. Uh We would do 9999 provided by to round down. That's going to give us 4,999. Yeah. Then we would take 1000 provided by two Round down though. That's going to be an integer. You have 500 now. We need to be careful here. It's not the number of even numbers between 9999 and 2000 Isn't just going to be 4999 -500 Because 1000 itself is within our interval as well as being an even number. So it would actually be 4999 Plus 1 -500 Or 5,000 -500. giving us 4500 for part C. We want to answer how many have distinct digits now for figuring out the number with distinct digits. We want to consider essentially having little boxes that were filling in with our numbers. The first box well we can't have 00 is not an option. We can only have between one and nine. So we have nine choices. Then for our second box actually I suppose Shouldn't put nine in that box. For our second box we can choose anything between zero and 9 except for what we put into the first box. So we have 10 choices -1. So we'll have nine again. Similar thing for the next box, we would have 10 choices -2. So 10 to 8 and then we have 10 choices minus three. So we have times 79 times nine times eight times seven is equal to 4536. Next we want to answer how many are not divisible by three. So it's going to be equal to 9000 The total number of positive integers. Just within. Actually it would be um one second. Yes. So there are 9000 positive integers between 1,009,999. Then oh we want to subtract off All of the integers less than 9,999 which are divisible by three. Yeah. And then we are going to need to add actually essentially add back in all of the integers less than 9000 or less than 1000 which are divisible by three. Essentially When we subtract off all of the integers less than 9,999 which are divisible by three will be taking away too many elements So do that. Hey. Yeah. Which is going to be the same thing as 9000 -3333 plus 300 in 33. Which is going to give us 6000 Now for the number that are divisible by five or 7. Keep in mind if we want to find out the number of things in set A. Or set B. That's equal to the number of things and said A plus the number of things in set B minus the things in the intersection of set A and B. So we want to find the numbers that are divisible by five, Add on the numbers that are divisible by seven. Then subtract off the numbers that are divisible by both. So for divisible by five would be mhm 9999 Over five rounds down minus 1000 over five. Which is going to give us 1,799. We will need to be careful here Because 1000 itself is divisible by five. It's well note that divisible by five invisibly visible by five. Yeah That number is going to be 1800 because we need to add back in 1000 A 1000 referring to the number. We need to add that back into our set of valid options. The size of the overall set is going to be 1800 Than the number divisible by seven. Similar deal. Except for we won't have to do that. Re accounting for the endpoint. We have 9999 divided by seven round down -1000 divided by seven. Round down Yeah. Which is equal to 1,286. And lastly we need to do to determine the number that are divisible by both. We do 9999 Divided by five times 7. Round down minus 1000, Divided by five times 7. Round down That equals 257. So visible the total number divisible by visible. Bye five or seven equal to 1800 minus or not minus. Excuse me, 1800 plus 12 86 -257. That is going to give us 1,543 as our total For part f. He wants to answer how many are not divisible by either five or 7. Excuse me? I need to jump back a second here because I just realized I wrote down the wrong number uh actually 18 plus 12 86 minus 2 57 is going to be 2829. Excuse me To determine the number that are not divisible by either five or 7. We just need to take the number which are the total number of integers between 1009 and 9999. And subtract off what we found in E So there are 9000 possible integers. We subtract off 2000 829 And that is going to give us 6171. Yeah. For part G. We want to find the number that are divisible by five but not by seven of that is just going to be 1000. So that would be we can figure it out by taking the number that are divisible by uh by five. Yeah. And subtracting off the numbers that are divisible by five and 7. So that would be 1800. That's the number divisible by five. And then we found that the number divisible by five and 7, it's 257. Yeah, So the number is 1543. And lastly our h this is actually something we figured out as part of trying to answer the previous question. Part E. So the number that are divisible by five and seven was 257.

In this question were asked if 148,000 and 70 is divisible by five or not to answer this. We look in the ones place as long as the ones place in a number has either a zero or a five than it is divisible by five. Our overall number is divisible by five, which means that our final answer here is 148,000 and 70 is divisible by five because we have a zero in the ones place.


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