Find the smallest positive integer of the set {306n : n € Z}n {657n :n € 2}...

The smallest positive integer $\mathrm{n}$ for which $\mathrm{n} !<\frac{(\mathrm{n}-1)^{\mathrm{n}}}{2}$ holds is (1) 4 (2) 3 (3) 2 (4) 1

In this problem of sex and counting, we have to list the elements in the given set and the set and this is we have to write the elements of set in Of all negative integers greater than -3. So when we write a number line so this is number line And here we have -3 And when we moved on the number line From a .2 right side Number increases or when we can say that the value increases. So this way we will remind the street, then it comes -2 then minus one, then zero, then one and then two and so on. And here also -3 -4 and so on. Here also 2345 and so on. The set end of all negative interiors Greater than -3. Let me is greater than minuses. This value will be -2 and -1. Only negative indigenous out there only minus two and minus one. So set and would be A set that has only two elements that are -2 and -1. So this is -2 and -1. So this is the required set.

In this question, we will learn about how to solve the maximum minimum problems with the help of differentiation. Right? The question is this expression given That is a fantasy equals to six and by four -16 and Cube Plus 9 & square. We need to find the value of and for which this function is minimum. Right? So the approach to solve this question is we are going to find the differentiation of dysfunction And we're going to create this 20. So let's find the first derivative of this function. So if the action would be equal to 24 and Cube -48 any square plus 18. And so this has to be created to judo to find for the minimum condition of the function. Therefore foreign, cube minus. Yeah, solid. For any square We can take six and out. So you get four and the square minus 16 and plus three is equal to zero. Right? For in the square minus eight. And let's say x equals to judo. So from here we get three values of N And can be idle judo and can be either one. Well then can be able to. Right? So one approaches we can find the value of the function on the history and values and we can choose when the function is minimum. Right? So we'll find the function value at an equal to zero. So that would be equal to zero. Late For n equals to one. This would be equal to six minus 16. Last night that is -1 and for n equals to do The value of function would be equal to six, multiplied by 2.4 -16, multiplied by two power 3 Plus nine, multiplied by two square, Which is equals to four. So from these three values, we can see that The function values minimum when N is equal to one Right, so the correct solution is and is equal to one.

Were asked to find the smallest positive integer with exactly end different positive factors. In part, a were given that N is equal to three. So defying the smallest such positive imager. Let's consider the factors of the positive managers until we find the energy or with exactly three factors. So on the left I'm going to write the factors, the integer and and on the right, all right. It's factors. So we have vinegar is one. This has factor one managers to says The two factors. One and two finishers. Three. This has the two factors one in three and if the integer is four four has yet factors one, two and four. So for is the smallest positive imagery, with exactly three different factors. So it follows that for part A. Our answer is four. So I'll circuit with us in Red Part B were given that N is four. So I want to find the smallest positive injure with exactly four different positive factors. So, looking back through our list, we saw that none of the Pasi majors, less than equal to four, have four factors. Even so, we have to keep going. Five has factors one in five. Six has the factors. 12 three and six. This is four factors, so it follows that six is the smallest positive integer with exactly for different factors. So this is our answer for part B in part C. Forgiven. The end is five. So you want to find the smallest positivity with exactly five different positive factors. So far, all the answers we considered have at most four positive factors. So we need to keep working. Seven has positive factors. 17 eight has positive factors. One to four and eight nine Has positive factors. One. Three in nine 10 as positive factors. One to five and 10 11 Has positive factors. One and 11 12. His positive factors. 123 four, six 12 So that's six factors right there. However, this is mawr than five factors. You want to find the smallest number with exactly five factors. So moving on we have 13 is positive factors. One in 13 14 has positive factors. 127 and 14 15 is positive factors. One, three, five and 15 16 has positive factors. One to four, eight and 16 These air five positive factors, so it follows that 16 is the smallest positive imager, with exactly five different positive factors. So this is our answer for part C for Part D. Were given that and is equal to six. So we're trying to find the smallest positive manager with exactly six different positive factors. We actually already encountered this one so recall that 12 had exactly six positive factors. So it follows that the answer to Parc de is 12 and finally in part E were given. That end is 10. So we're skipping along here rest to find the smallest positive injure with exactly 10 different positive factors. So this might take a bit longer. So consider 17 this positive factors 1 17 18 is positive factors. 12 three, six, nine and 18 19 is positive factors. One of 19 2 factors, 20 has factors. 12 four, five, 10 20 So six positive factors 21 has thief actors 13 seven 21 4 Positive factors 22 Has factors 12 11 22 four Factors 23 has factors 1 23 two Factors 24 has factors one to three four six eight 12 24 So eight factors We're getting close, but we're not there yet. 25 as one five 25. This factors three factors. 26 has the factors one to 13 26 four factors 27 Has factors one, three nine. 27 So four factors 28 as factors one to four seven 14 28 So six Factors 29 has factors one and 29 two Factors 30 Has factors one to three, five, six 10. 15 30 Again eight factors So we're getting close. 31 Has factors one in 31 two factors 32 as factors one to four eight 16 32 six Factors. 33 as factors. One, three, 11. 33 So four factors. 34 as factors. One to 17 and 34 So four factors 35 has factors one five seven. 35 So four factors 36 has factors one to three, four six nine. 12 18 36 So nine factors or even closer now need one more factor. 37 as factors. One in 37 So two factors 38 has factors one to 19 38 So four factors 39 has factors. One 3 13 39. So four factors 40 These factors one to four, five, eight, 10 2040 So eight factors Not quite there. 41 has factors one and 41 So two factors 42 has factors. One to three, 67 14 21 42 Eight Factors again close but we're two away. 43 Has factors 1. 43 two Factors 44 Has factors 12 four, 11 22 4 Before So six factors 45 as factors. One, three five nine. 15 in 45. So six factors 46 as factors. One to 33. That was a mistake. Not 33 23 and 46. So four Factors 47 has factors. One in 47 two factors. Mhm 48 has factors one to three, four six eight 12 16 24 48 for a total of 10 factors. So it follows that the smallest positive integer with exactly 10 different positive factors is 48

Hello. So first here we just recall that the integers are just the set of all the natural numbers and they're opposites. So we go you know the natural numbers. There's 12345 And then the opposite you know native one. Native to. Native to Native 45 so on. And then also um zero is an integer. There's all the integers. And we asked our here what's the smallest positive integer or to be a positive integer? We have the greater than zero. So the smallest number, that's the smallest integer. Whole number that's greater than zero is one. So the smallest positive integer. Um Well this is pretty simple right? That is just one uh by the density property. Um There are infinite there are infinitely many rational numbers and rational numbers um between any two distinct real numbers. So therefore we can go ahead and conclude that there is no such thing as the smallest possible rational number and the smallest um positive positive irrational number. So no smallest um no smallest rational um or irrational. Thanks. Right because again between any two real numbers there is a rational number and between any two real numbers there is an irrational number. Um So you can go as small as you want, you're always gonna be to find another one. So there's gonna be no smallest rational or irrational number. Um And um and yes that when we asked the um let's see here um the smallest positive integer was the smallest positive rational spots irrational. So yeah that basically um answers the question. So yes I guess we're done all right.