1 00:00:00,990 --> 00:00:06,810 It's kind of a geeky maths question, but it's actually pretty nice. 2 00:00:07,610 --> 00:00:14,160 No, I think Hilbert's hotel is from something like Hilbert's curve or some function. 3 00:00:14,180 --> 00:00:21,630 And I don't know this guy Hilbert was some math guy who I think had some fury about the infinite numbers 4 00:00:21,630 --> 00:00:22,140 or something. 5 00:00:22,710 --> 00:00:25,140 But anyways, let's read the statement. 6 00:00:26,070 --> 00:00:33,660 So as you probably know, you are given this hotel and basically you have an infinite number of rooms 7 00:00:33,660 --> 00:00:35,820 from minus infinite to plus infinite. 8 00:00:35,850 --> 00:00:43,350 And there is exactly one person staying in each of these rooms called everything and not normal for 9 00:00:43,350 --> 00:00:44,400 a hotel right now. 10 00:00:44,760 --> 00:00:49,870 But the interesting thing is that you are given an array. 11 00:00:50,350 --> 00:00:50,530 Right. 12 00:00:51,550 --> 00:00:53,740 Just called a and has an integers. 13 00:00:54,190 --> 00:00:57,510 And what happens is that for each number key. 14 00:00:57,550 --> 00:01:00,340 So for each room, which has no key. 15 00:01:00,760 --> 00:01:02,770 Of course from minus infinity. 16 00:01:02,800 --> 00:01:03,550 Plus infinite. 17 00:01:04,690 --> 00:01:11,410 The guy who is staying in that room, number K, is now going to be asked to leave that room and go 18 00:01:11,470 --> 00:01:17,120 to room number K plus A. of K modulo N. 19 00:01:17,600 --> 00:01:19,900 And this module is. 20 00:01:20,950 --> 00:01:21,180 Yeah. 21 00:01:21,730 --> 00:01:24,850 They give us some definition for this module, but we are gonna get into it. 22 00:01:25,330 --> 00:01:28,990 So let's see how I thought about this problem. 23 00:01:29,200 --> 00:01:30,940 So let's take some examples first. 24 00:01:30,950 --> 00:01:31,170 Right. 25 00:01:31,450 --> 00:01:33,070 I have basically read this problem. 26 00:01:33,160 --> 00:01:36,970 And of course, first thing I know, it's a math stuff. 27 00:01:38,020 --> 00:01:42,370 And basically, when you're reading on good force is a problem that's about math. 28 00:01:42,940 --> 00:01:46,510 You usually have to get your pen and paper close to you. 29 00:01:47,140 --> 00:01:47,980 I didn't do that. 30 00:01:48,190 --> 00:01:51,640 I took my risk and it turned out I can solve it. 31 00:01:51,640 --> 00:01:54,040 Like only thinking about it, not writing anything. 32 00:01:54,460 --> 00:01:55,470 So let's see. 33 00:01:55,840 --> 00:01:56,770 Let's take some examples. 34 00:01:57,070 --> 00:02:05,590 What I did is I go look at these examples and let's take the first one, which is quite interesting. 35 00:02:05,980 --> 00:02:08,770 So we have four elements, right? 36 00:02:09,520 --> 00:02:11,830 We are given four elements. 37 00:02:12,090 --> 00:02:17,580 Four, they are not the rooms either or for the four elements and equals four. 38 00:02:17,740 --> 00:02:18,070 Whatever. 39 00:02:18,490 --> 00:02:19,430 So any plans for. 40 00:02:19,900 --> 00:02:23,320 And the array A is by. 41 00:02:26,990 --> 00:02:28,990 I've and I think the last element was what? 42 00:02:30,410 --> 00:02:30,740 Sure. 43 00:02:30,950 --> 00:02:31,340 Was it? 44 00:02:31,640 --> 00:02:34,730 Yes, it was one, so that's what happens here. 45 00:02:34,940 --> 00:02:38,020 First of all, let's index these integers, right? 46 00:02:38,270 --> 00:02:41,770 Zero, one, two and three. 47 00:02:42,820 --> 00:02:47,680 And let's try to imagine the rooms of our. 48 00:02:49,400 --> 00:02:55,810 To have here it's I'm going to use my drawing skills, so minus infinity. 49 00:02:59,140 --> 00:03:01,600 Minus infinite, plus one. 50 00:03:02,350 --> 00:03:03,850 And so on and so forth. 51 00:03:07,100 --> 00:03:07,940 Number zero. 52 00:03:11,700 --> 00:03:14,350 Room number one and so on and so forth. 53 00:03:15,330 --> 00:03:17,550 And they'll, of course. 54 00:03:18,740 --> 00:03:19,610 Room number infinite. 55 00:03:20,300 --> 00:03:24,530 And don't ask me how I know this is room number infinite because I don't. 56 00:03:24,800 --> 00:03:30,920 But let's imagine we can get in front of room number infinity or minus infinity, right? 57 00:03:31,130 --> 00:03:32,780 If we had superpowers or something. 58 00:03:33,200 --> 00:03:36,830 So basically, let's imagine all the rooms in our hotel. 59 00:03:36,890 --> 00:03:37,190 Right. 60 00:03:38,080 --> 00:03:39,680 Binding the roll call. 61 00:03:40,260 --> 00:03:41,470 And let's see what happens. 62 00:03:42,070 --> 00:03:51,460 Well, as we said, the guy in rule number case is asked to leave the room and move to Kate plus eight. 63 00:03:52,700 --> 00:03:57,920 Of I think it was the AK modulo and. 64 00:03:59,290 --> 00:03:59,580 Cool. 65 00:04:00,240 --> 00:04:02,610 So let's see what they did here. 66 00:04:04,470 --> 00:04:09,960 I have taken a look at this formula and I see. 67 00:04:11,610 --> 00:04:14,540 Free, let's say, letters. 68 00:04:15,630 --> 00:04:16,290 We just keep. 69 00:04:18,140 --> 00:04:19,740 And and cool. 70 00:04:20,850 --> 00:04:25,710 So they basically see here is some integer key. 71 00:04:27,880 --> 00:04:42,160 Which is between minus infinite and infinite and some integer A of K modulo M, which from, of course, 72 00:04:42,250 --> 00:04:43,260 the input array. 73 00:04:44,290 --> 00:04:45,430 So I'm thinking like this. 74 00:04:45,850 --> 00:04:48,460 All right, man, I have two integers. 75 00:04:49,390 --> 00:04:51,070 One of them is infinite. 76 00:04:51,090 --> 00:04:55,500 Could be anything I can possibly fix the key or something like that. 77 00:04:55,510 --> 00:04:55,780 Right. 78 00:04:56,710 --> 00:05:00,130 So I really have to. 79 00:05:01,700 --> 00:05:02,760 Do something with this. 80 00:05:04,260 --> 00:05:06,690 But what they basically did is that. 81 00:05:08,720 --> 00:05:09,620 Because Kee. 82 00:05:11,020 --> 00:05:12,520 Can take an infinite. 83 00:05:13,490 --> 00:05:14,960 Of possible values. 84 00:05:18,290 --> 00:05:20,030 Eight of Kay modulo and. 85 00:05:21,000 --> 00:05:23,340 Take only impossible values. 86 00:05:23,520 --> 00:05:23,820 Right. 87 00:05:24,000 --> 00:05:25,530 Because there are values in disarray. 88 00:05:26,310 --> 00:05:26,940 So there are. 89 00:05:27,030 --> 00:05:28,610 And possible. 90 00:05:31,310 --> 00:05:36,570 Boy, did I do I didn't fix Kay because no one can do that. 91 00:05:37,140 --> 00:05:40,140 But I have fixed Kay modulo. 92 00:05:42,480 --> 00:05:48,510 I have fixed each and every key module and of course. 93 00:05:49,680 --> 00:05:52,170 This is between zero and minus one. 94 00:05:53,620 --> 00:05:55,410 That's the first thing I know I have to do. 95 00:05:57,300 --> 00:05:59,460 And let's start from the beginning. 96 00:06:01,400 --> 00:06:06,500 Let's take the case in which key module N is equal to zero. 97 00:06:08,390 --> 00:06:11,840 Let me erase this down here and see what happens. 98 00:06:13,710 --> 00:06:17,620 What's going to happen on Teneriffe? 99 00:06:17,720 --> 00:06:18,330 Number one. 100 00:06:19,500 --> 00:06:21,420 Kay Modula. 101 00:06:26,200 --> 00:06:32,320 Well, first of all, let's take a look at a zero, which is five. 102 00:06:36,020 --> 00:06:37,460 What does the problem tell us? 103 00:06:39,290 --> 00:06:43,670 That the guy in each and every no key. 104 00:06:45,870 --> 00:06:48,370 Which is now a multiple of em. 105 00:06:49,230 --> 00:06:49,590 Right. 106 00:06:49,950 --> 00:06:52,890 Because we have set this reminder, we know it right now. 107 00:06:53,880 --> 00:06:55,890 So what are the possible values? 108 00:07:01,200 --> 00:07:03,780 Which is this room then? 109 00:07:06,350 --> 00:07:07,100 Could be an. 110 00:07:10,700 --> 00:07:11,390 Let's say here. 111 00:07:12,650 --> 00:07:13,310 We have room. 112 00:07:14,590 --> 00:07:15,280 No n. 113 00:07:16,460 --> 00:07:18,380 Could be cross then. 114 00:07:21,520 --> 00:07:22,240 We have room. 115 00:07:23,700 --> 00:07:24,630 Number two, cross that. 116 00:07:25,670 --> 00:07:28,310 So on and so forth, of course, infinite Vellis here. 117 00:07:29,520 --> 00:07:32,790 And also, there are negative versions. 118 00:07:32,970 --> 00:07:33,270 Right. 119 00:07:33,870 --> 00:07:35,650 Which is minus two crossed. 120 00:07:37,160 --> 00:07:38,540 Minus four crust and. 121 00:07:42,100 --> 00:07:43,750 So here I have. 122 00:07:44,690 --> 00:07:45,590 Some doors. 123 00:07:45,620 --> 00:07:46,370 Some rooms. 124 00:07:49,230 --> 00:07:51,900 Minus two, cross that and so on and so forth. 125 00:07:54,140 --> 00:08:04,430 What happens is that I can imagine that all these guys are taken out of these rooms and they are shifted 126 00:08:04,730 --> 00:08:05,330 by five. 127 00:08:07,410 --> 00:08:11,220 That's all they are, just shifted by five. 128 00:08:16,470 --> 00:08:18,540 Bus five and so on and so forth. 129 00:08:21,460 --> 00:08:23,170 This hotel is infinite. 130 00:08:27,740 --> 00:08:32,960 There is actually one thing which we can mathematically notice about these numbers. 131 00:08:34,940 --> 00:08:43,760 What happens is that we kind of rebind these numbers what used to be zero. 132 00:08:45,540 --> 00:08:45,880 Is No. 133 00:08:45,880 --> 00:08:51,600 Five, what used to be N is now and plus five and so on and so forth. 134 00:08:53,260 --> 00:09:05,290 So what I did is the following thing, if I can imagine an array of infinite numbers from minus infinite 135 00:09:05,560 --> 00:09:14,020 to infinite and every value in disarray is one, because there is one guy there on that position. 136 00:09:17,840 --> 00:09:20,600 Which we will call Geary of rooms. 137 00:09:20,900 --> 00:09:21,200 All right. 138 00:09:21,930 --> 00:09:23,410 Well, the array of rooms. 139 00:09:30,700 --> 00:09:36,210 Plus Symphonie, plus one dot, dot, dot zero, one dot, dot, dot. 140 00:09:36,950 --> 00:09:37,080 If. 141 00:09:41,420 --> 00:09:52,520 On this or in disarray is one when Kay modulo n e0, I literally take all these positions. 142 00:09:53,950 --> 00:09:54,670 Position zero. 143 00:09:56,980 --> 00:09:59,570 Xitian minus N position to cross N. 144 00:09:59,910 --> 00:10:02,150 Position minus two crossed in somewhere here. 145 00:10:02,610 --> 00:10:04,750 And all I do is take. 146 00:10:05,880 --> 00:10:07,280 That one here. 147 00:10:08,490 --> 00:10:09,560 And the movie it. 148 00:10:11,930 --> 00:10:14,920 But that position plus five for this example. 149 00:10:17,110 --> 00:10:19,530 So I basically asked myself the following thing. 150 00:10:21,570 --> 00:10:28,280 How can I make sure or how can I check if after performing all these moves basically. 151 00:10:28,760 --> 00:10:29,090 Right. 152 00:10:29,460 --> 00:10:33,180 Subtracting one from here and increasing one here? 153 00:10:33,360 --> 00:10:33,720 Yes. 154 00:10:34,410 --> 00:10:36,660 How can I make sure that the new array. 155 00:10:38,500 --> 00:10:39,130 Let's call it. 156 00:10:39,880 --> 00:10:41,750 Those are free of those, actually. 157 00:10:42,180 --> 00:10:42,810 Let's call it a. 158 00:10:45,830 --> 00:10:50,540 The new array room's prime also has only values of one. 159 00:10:53,160 --> 00:10:55,130 They have to make sure, oh, this. 160 00:10:56,310 --> 00:11:03,030 So basically, I have to make sure that there isn't any position he. 161 00:11:06,300 --> 00:11:08,070 Where there are two. 162 00:11:13,120 --> 00:11:14,740 Positions X and Y. 163 00:11:16,020 --> 00:11:17,670 That are going to be. 164 00:11:20,800 --> 00:11:22,020 Basically to get here. 165 00:11:24,460 --> 00:11:34,900 So I have to make sure that there are in two different rooms, X and Y, that are going to be binded 166 00:11:34,930 --> 00:11:35,950 to the same number. 167 00:11:36,910 --> 00:11:42,050 Basically, I have to make sure that this function, which takes my people from their rooms and move 168 00:11:42,050 --> 00:11:44,770 it to other rooms, is how do you call it? 169 00:11:48,010 --> 00:11:50,950 I can think about the term by objective or something. 170 00:11:51,010 --> 00:11:52,290 I think that's the term for it. 171 00:11:52,290 --> 00:11:52,400 Yes. 172 00:11:53,260 --> 00:11:58,050 So there are not any two different values which are going to be equal. 173 00:12:00,700 --> 00:12:01,690 Having thought about all this. 174 00:12:02,560 --> 00:12:03,850 Get back to this K module. 175 00:12:04,020 --> 00:12:06,740 And now everything is gonna be clear. 176 00:12:09,890 --> 00:12:14,630 This example on this case, on this scenario when Kay modulo any zero. 177 00:12:17,330 --> 00:12:21,320 What happens is that each number. 178 00:12:22,270 --> 00:12:26,290 Each position from this array, which is a multiple of an. 179 00:12:29,300 --> 00:12:32,690 Now is not going to be any more a multiple of an. 180 00:12:33,990 --> 00:12:36,930 But it's a reminder modulo and is going to be five. 181 00:12:40,400 --> 00:12:41,640 Port, this scene aerial. 182 00:12:44,010 --> 00:12:45,800 Where let's say here. 183 00:12:47,500 --> 00:12:47,800 Right. 184 00:12:48,190 --> 00:12:49,650 Kay modulo n zero. 185 00:12:51,870 --> 00:12:52,470 Basically. 186 00:12:53,950 --> 00:12:59,350 Each and every number, which modulo N was zero. 187 00:12:59,950 --> 00:13:00,280 Right. 188 00:13:00,340 --> 00:13:03,700 So it was a multiple of N No. 189 00:13:05,000 --> 00:13:10,610 From multiple of end, let's say something like, I know D crust n. 190 00:13:11,570 --> 00:13:13,540 Is now the crust. 191 00:13:13,840 --> 00:13:18,070 And plus a of zero, which is this value. 192 00:13:19,790 --> 00:13:22,670 So if that number. 193 00:13:25,540 --> 00:13:26,630 Modulo an. 194 00:13:28,880 --> 00:13:30,110 Was zero. 195 00:13:31,900 --> 00:13:33,610 After performing this. 196 00:13:35,880 --> 00:13:38,100 The new number modulo an. 197 00:13:41,520 --> 00:13:42,480 To be a zero. 198 00:13:43,670 --> 00:13:46,800 Well, a zero, of course, modulo an. 199 00:13:48,840 --> 00:13:51,210 So the whole thing that happens here. 200 00:13:54,970 --> 00:13:57,070 One reminder becomes another. 201 00:13:58,620 --> 00:14:02,660 Why do we have to imagine instead of that infinite array of rooms? 202 00:14:03,630 --> 00:14:11,850 Is that these rooms are grouped by the reminder of their index, their number modulo N. 203 00:14:12,660 --> 00:14:12,960 Right. 204 00:14:13,920 --> 00:14:19,800 So on that infinity of numbers, I'm going to have the numbers which modulo N. 205 00:14:20,840 --> 00:14:26,190 Reminder zero, the ones which give reminder, one reminder, too. 206 00:14:26,990 --> 00:14:28,310 And so on and so forth. 207 00:14:29,280 --> 00:14:37,480 Lastly, all the numbers from minus infinity infinite, which modulo n give reminder and minus one. 208 00:14:38,800 --> 00:14:43,060 Of course, there are an infinity of numbers in each of these and groups. 209 00:14:43,480 --> 00:14:44,560 But that doesn't matter. 210 00:14:47,780 --> 00:14:50,000 When I took a of zero. 211 00:14:52,330 --> 00:15:00,300 Each and every number of reminder zero became now a number of reminder, a zero modulo an. 212 00:15:01,320 --> 00:15:07,740 This zero becomes a zero modulo and. 213 00:15:11,190 --> 00:15:13,110 Then what about reminder one? 214 00:15:13,860 --> 00:15:15,690 Well, each number. 215 00:15:17,570 --> 00:15:19,310 Which was. 216 00:15:23,880 --> 00:15:26,100 Something like this right here one. 217 00:15:27,600 --> 00:15:29,640 Each number, which was something like decreased. 218 00:15:29,710 --> 00:15:37,260 And plus what gave reminder one now is going to be increased by a one. 219 00:15:37,950 --> 00:15:43,680 So is D crust and plus a of one plus one. 220 00:15:45,400 --> 00:15:47,980 This reminder was. 221 00:15:51,550 --> 00:15:52,490 A new reminder. 222 00:15:55,450 --> 00:15:59,140 A one, us one and everything. 223 00:16:00,220 --> 00:16:01,380 Module at. 224 00:16:07,190 --> 00:16:08,960 But to what happened? 225 00:16:09,710 --> 00:16:10,520 Exactly the same thing. 226 00:16:13,030 --> 00:16:14,380 So reminder to. 227 00:16:17,720 --> 00:16:19,990 A of two plus two modularly. 228 00:16:21,380 --> 00:16:22,180 And you've got a whole idea. 229 00:16:25,510 --> 00:16:28,390 So what happens is that. 230 00:16:30,190 --> 00:16:32,590 For each and every reminder. 231 00:16:35,370 --> 00:16:37,320 Between zero. 232 00:16:43,130 --> 00:16:49,310 This is going to be transformed, as you've seen, into air plus eight of our. 233 00:16:50,240 --> 00:16:51,470 Everything Modula And. 234 00:16:58,280 --> 00:17:00,710 Did we know this or what? 235 00:17:04,620 --> 00:17:08,100 What did we agree that we will have to check? 236 00:17:09,210 --> 00:17:10,440 Remember we said. 237 00:17:14,980 --> 00:17:22,180 There shouldn't be two rooms X different than Y such that after the transformation. 238 00:17:27,950 --> 00:17:34,760 And as we've seen, because we transform the numbers grouped by reminders. 239 00:17:36,050 --> 00:17:37,200 This can happen. 240 00:17:39,710 --> 00:17:41,060 If and only if. 241 00:17:42,280 --> 00:17:44,450 The reminder of X modulo eight. 242 00:17:46,930 --> 00:17:48,940 A reminder of why modulo. 243 00:17:50,620 --> 00:17:51,800 If these two are different. 244 00:17:51,820 --> 00:17:57,160 Of course, they're reminders are going to be different if this two reminders. 245 00:17:58,340 --> 00:17:59,630 After applying. 246 00:18:00,570 --> 00:18:03,480 The transformation for all the reminders. 247 00:18:04,320 --> 00:18:05,520 Become now. 248 00:18:07,050 --> 00:18:07,800 Same reminder. 249 00:18:08,280 --> 00:18:10,490 Which is Ze modulo n. 250 00:18:11,590 --> 00:18:12,780 So all we have to do. 251 00:18:15,800 --> 00:18:23,150 Each of these numbers are plus eight of our modulo n are also going to be. 252 00:18:24,080 --> 00:18:24,740 Different. 253 00:18:25,700 --> 00:18:28,040 All the values from zero to N minus one. 254 00:18:28,460 --> 00:18:28,760 Right. 255 00:18:29,150 --> 00:18:33,560 So we have basically started from the. 256 00:18:35,790 --> 00:18:41,010 Permutation of all the numbers from zero to minus one in this order. 257 00:18:41,940 --> 00:18:43,470 And we will update. 258 00:18:44,530 --> 00:18:45,580 Another permutation. 259 00:18:47,780 --> 00:18:52,000 Based on the Formula R plus eight of our Modulo n. 260 00:18:56,990 --> 00:19:03,030 Typically, we will obtain an array, and if this array is a permutation that everything is fine because 261 00:19:03,060 --> 00:19:05,880 as I've told you, if this is a permutation. 262 00:19:06,990 --> 00:19:10,410 This means that each two different reminders. 263 00:19:11,280 --> 00:19:14,430 After the transformation are still different. 264 00:19:16,320 --> 00:19:16,650 Right. 265 00:19:17,070 --> 00:19:18,690 So if our one. 266 00:19:19,610 --> 00:19:20,630 And ah, to. 267 00:19:21,530 --> 00:19:27,180 Or different from this permutation and after the transformation, they are still different. 268 00:19:30,060 --> 00:19:31,920 This means that those numbers. 269 00:19:33,450 --> 00:19:34,380 X and Y. 270 00:19:36,230 --> 00:19:41,540 Which were X in group air one and Y in group F two. 271 00:19:44,130 --> 00:19:46,800 What happens after they are transformed? 272 00:19:47,790 --> 00:19:50,280 They're reminders modulo and are different. 273 00:19:50,310 --> 00:19:53,440 Of course, the numbers are going to be all. 274 00:19:53,850 --> 00:19:54,250 Also the. 275 00:19:55,700 --> 00:19:56,210 And that's it. 276 00:19:56,990 --> 00:19:59,090 This is how you solve this problem. 277 00:20:01,400 --> 00:20:11,420 You don't think about the infinite possible values of the room numbers, but you think about their fine, 278 00:20:11,430 --> 00:20:14,910 neat posable reminders modulo and. 279 00:20:15,260 --> 00:20:17,540 And in this way, you have something. 280 00:20:17,820 --> 00:20:18,590 Find it right. 281 00:20:18,980 --> 00:20:21,680 So you can iterate for those elements. 282 00:20:22,880 --> 00:20:24,320 So what they basically did here. 283 00:20:25,370 --> 00:20:26,750 The same thing is the following. 284 00:20:28,230 --> 00:20:31,250 We've basically have iterated through the whole array A. 285 00:20:33,100 --> 00:20:35,770 And I had something like a hash table. 286 00:20:37,540 --> 00:20:37,890 Stable. 287 00:20:38,710 --> 00:20:44,050 Unordered map, an order set frequency array vector from Estill. 288 00:20:44,230 --> 00:20:45,220 It's really your choice. 289 00:20:46,720 --> 00:20:49,630 Let's say it's a harsh day just for the sake of it. 290 00:20:50,200 --> 00:20:51,280 I've got the hash table. 291 00:20:52,750 --> 00:21:00,470 And for each and every element from disarray, I simply checked if that element A of all. 292 00:21:05,430 --> 00:21:06,150 Module An. 293 00:21:12,450 --> 00:21:13,620 Hasn't been encountered. 294 00:21:17,090 --> 00:21:25,580 If this value is not part of my hashtag, if it's not, that's cool, inserted into the hash table and 295 00:21:25,580 --> 00:21:27,680 continue if it is part. 296 00:21:29,880 --> 00:21:30,860 Timperley, return minus. 297 00:21:32,040 --> 00:21:32,580 And that was it. 298 00:21:34,350 --> 00:21:35,190 Just one thing. 299 00:21:35,490 --> 00:21:38,760 The problem was when these numbers. 300 00:21:42,250 --> 00:21:49,480 Because if these numbers are negative, the value AA plus I everything modulo N is not going to be correctly 301 00:21:49,510 --> 00:21:50,080 computed. 302 00:21:51,340 --> 00:22:00,070 What I did, basically, is that if this number is negative, I have increased it by a value like 10 303 00:22:00,100 --> 00:22:02,020 at power line, crossed an. 304 00:22:02,990 --> 00:22:08,570 So I make sure that this final value is positive and also. 305 00:22:09,890 --> 00:22:15,320 The reminder of this value is equal to the reminder of this value modulo. 306 00:22:16,280 --> 00:22:16,580 Right. 307 00:22:16,940 --> 00:22:20,810 Be very careful here because if you take something like minus two. 308 00:22:23,080 --> 00:22:23,860 Modulo 10. 309 00:22:25,640 --> 00:22:31,700 You should get eight for this problem, but you will get minus two, which is a problem. 310 00:22:32,390 --> 00:22:38,630 But if you take this minus two and increase it by something big enough, this becomes eight, which 311 00:22:38,630 --> 00:22:39,800 is greater than zero. 312 00:22:40,490 --> 00:22:44,720 And the reminder modulo 10 is eight, which is indeed the great. 313 00:22:45,880 --> 00:22:51,010 So the problem that I had and I sold it quickly, is to increase this number by something big enough 314 00:22:51,010 --> 00:22:52,010 so it becomes post. 315 00:22:53,070 --> 00:22:55,020 That's it all about the reminders.