All solutions should be clearly presented with complete justification _ Find prime factorizations for each of the following numbers_ 6" x 122 b) 10100 1098...

Find the prime factorization of each of these integers.
$$\begin{array}{ll}{\text { a) } 88} & {\text { b) } 126} & {\text { c) } 729} \\ {\text { d) } 1001} & {\text { e) } 1111} & {\text { f) } 909,090}\end{array}$$

So here in this question I have here is to the par four minus beers to depart for It can be done us a square holes where? Minus b square hold square, which I converted Us. Yes, we're minus P square, a squared plus B squared. If I observed this expert minus B squared, I can. For the writer, that's a minus bi a plus B. And then the remaining factor is 12 plus B squared, which cannot be factored anymore. So this is an answer. Similarly, here's to the parsecs finest leaders to the bar. Six. I can try to dust a que holds minus big. Your balls were so I contacted us a U minus vehicle in the a que plus vehicle so that can be read antagonists a minus bi in tow. Yes, where plus a B plus B squared. Then this can bitterness it. Let's be multiplier way s clip minus a B that's be square. So that can be simply 500 us a minus. Bi indoor air plus me multiple airway. Yes. Where? Let's be square less a B multiplied, right. A squared plus B squared minus a B. So that is my answer. So containing forward with the same cushion. I have to verify this expression 12 hours to the bar for minus seven rest of the power for so using the above the right formula, I will write it directly. Well minus seven in tow. 12 plus seven. I'm using this formula physically multiplied by 12 square last seven square. So this can be done. Us well minus seven is by 12 plus seven is 19 and this is 1 44 plus 49. So that value is equal to 1 93 No. 1 93 I have to multiply with 19 on five. So when I multiply this, I'll get the required result that this given to me as if this 19,335. Similarly, in order to do the next portion 12 x to the power six minus seven, restore the power six. I will use this derived formula A power six miners be power six. I have like this so I'll write down this one directly. 12 minus six. So? So this is going to be 12 minus seven and then next witness. Well, bless seven and then a squared plus B squared that this 12 square plus seven square plus 12 into 70 84 next one s 12 square plus seven square minus eight before, So that if I see 12 minus seven s five multiplied when 19 multiplied by this value ihsaa 1 93 that we have seen. So this is 1 93 plus 84. If I do, that will be to 77 on 1 93 minus before when I do that this under an it for multiplying all these values I'll be getting exactly the given result that this do it. Six it 335 So that is my answer. So this where Lewis 109 on Dr Moore delegation will get me the same result. Now, if I say that hard part that asking me of the results and party and we two factories, the Anteaters Okay, so now I have to fact right, this interiors he had been 1003 135. So in orderto tries it, I'll follow the same previous step. Andi, I'll be able to see that this is 12 years to the power for minor seven resto depart for so directly using calculator. I can get this result or from this step, I can go back. Fact rising like this and I can get back the same result similarly. Ah, this to 868335 can be returned as a product off these four numbers with our all co prime numbers on Ultimately that will need may toe this expression 12 x to the power six minus seven years to the power six. So I hope it's clear. Thank you.

We're being asked to find the prime factory ization of 144 using any method so you can even do a factor tree or you can use the ladder method. I'm going to use the ah factor tree, so I have to start by finding two factors that multiply the 144. Well, I know that 12 times 12 is 144 so I can break this up to 12 times 12. So let's start by looking at the 1st 12 Well, most applies to 12. Well, three times four is equal to 12 now. Three is a prime number. So that ends. This branch for is not prime. No, but I know that two times two is equal before now to his prime, which closes this bridge. And we just mentioned to his prime which closes this French. So now let's go back to that number 12. Well, again, we already mentioned three times four is 12 but three is prime, which closes this French. And we know that two times two is equal before and again to his prime, which closes both of these branches. So now I'm going to write the product of my circle values. And I'm also gonna go in order from least agreed this. So I'm gonna have two times. Two times, Two times, Two times. Three times three. Now, I'm just going to rewrite this an exponential for So if you know this, we have two times, two times, two times two, which can be rewritten as two to the fourth. Then we have three times three, which could be rewritten as three squared. So we've found that the prime factory ization of 144 is two to the fourth times, three square.

The first given number. He's that d nine it is equal to today in tow. Slept in next twenties 81 81 equals nine into nine. And again, Name Can beat it in as three into three is equal to three to the power off for next one is we're not one. We know that we're not one easier prime number. The next number is 1 43 is equal to 11 in two 13. We know that 11 is prime number and 13 is also a prime number. 2 89 2 89 conveyed it in as 17 into 17. We know that 17 easier prime number No. 17 in 2 17 equals to 17 square and eight 99. Hate 99 can be written as 29 in two. That the one we know that 21 easier time number and that the one is also a prime number. Thank you for watching

Let's write out a couple of different number sets, starting with the 1st 10 whole numbers. That's not so bad. The whole numbers just start with once we have 123 for five, 6789 10. Easy enough. Next up, we have the 1st 10 prime numbers. These are a little bit more interesting. They're still not so bad. One is a prime number. Ah, then we have two, which is the only even prime three. Four is not a crime five. And you'll notice there will be no more evens from here on out. Seven on nine is doing well by three, but 11 is prime. Then there's 13 17 19. Let's see how many primes is this were up to nine primes. We just need one more that is, well, not 21 not 22 but 23. And that is the 1st 10 primes. Now you may encounter some people who say that one and two do not count as primes. I personally am not one of those people. One and two are in fact prime numbers. However, if you encounter something like this, then you could throw up a couple of more primes, that is, 29 and 31 are the next two primes in order, so you shouldn't need them. And finally, the set of the integers. This one's not as straightforward because you can't just name them. That is their infinite integers. If you want to name them all, you'd be here forever. So we'll just name something that tells you what they look like. That is negative. Four negative three negative to negative. 10123 and so on. Every whole number, positive or negative, is an integer. Thus this is the set of the inter germs and therefore we have found all of the sets that this problem asked us to find.