00:01
All right, so first of all, notice that when you've got a number n, you can always write it as n equals 100a plus b.
00:09
I mean, a can even take the value zero for the first values, but you can always write it in this form here.
00:15
The reason i'm saying this is that n squared, so is equal to 100a plus b squared, which is 10 ,000 a squared plus 200 ab, plus b squared.
00:33
Now notice that this number here will always have the last two digits being equal to zero zero, all right? because you've got this 10 ,000 here.
00:44
And now here again, since you've got 200, the last two digits of this product here will also be zero zero.
00:52
So what we've just shown here is that you only have to consider the the b basically of the if you decompose n in this shape where let me just precise that b is between 0 and 99 well by checking only the last two digits of n well you can know the last two digits of n squared and that's already a big step because if we're trying to show something for all n we're basically trying to show it for an infinite number of values um which is is not something possible in if you want to prove something by exhaustion or proof something by being doing a proof by case and now we've reduced it to 100 different cases so one way of doing it would be simply to check for all the different values of b b ranging from 0 to 99 of b square sorry where b ranges from 0 to 99 and conclude but that is obviously very uh takes of time um and it's quite boring.
02:04
So what you want to do is to reduce the number of cases to check.
02:10
And by further, you want to further reduce the number of cases to check.
02:17
Now, why exactly did i straight, jumped straight into this decomposition? well, intuitively, you got to tell yourself that, you know, if you've got a number like 6803 ,000, 5 ,000.
02:34
440, that this part here, when it is squared, it doesn't really contribute that much to the last two digits of the number when it's squared.
02:48
Because these, i mean, this is a fuzzy argument again, but this is often what leads you to some form of more rigorous proof.
02:58
Okay.
03:00
All right.
03:00
So that was a short parenthesis.
03:03
So how do you reduce the number of cases? to check once you've established that you only have to look at b ranging from 0 to 99.
03:11
Well, the first thing is to notice that 100 minus b squared, right, is equal to 100 squared minus 200b plus b squared.
03:28
And since you only care about the last two digits, well, again, you notice that this ends in 0, this ends again in zero zero by the same argument we've made before and therefore the last two digits of 100 minus b squared is the same as the last two digits of b squared so what does that tell us well basically we've reduced the number of cases to check by two we've divided by the number of cases by two why? well, basically what this means is that checking the last two digits of a number, let's say four, is the same as a four square, sorry, is the same as checking the last two digits of 100 minus four squared, which is 96.
04:18
All right? 96 squared, sorry.
04:22
And, you know, if you check this, four squared is equal to 16 and 96 squared.
04:28
Is 9 ,260.
04:32
So you know, you see that both end in 60.
04:35
Okay.
04:36
And why do i say 50? well, i mean, you can see that.
04:40
If, you know, you start with b equals 1, then that tells you that 99 has the same, 99 squared has the same last digits as 1 squared.
04:55
Then you'd say 2, same as 98, and you go down, go down, go down, and you've basically divided, oh sorry, you've divided those 100 numbers in two halves where you've got 50 on both sides.
05:09
Okay.
05:10
Now that's already very good.
05:12
You divide the number of case by two, but you can go one step further.
05:16
And the thing is here, if you notice here, that two is, sorry, that two here is almost too big.
05:28
It's not necessary.
05:28
You could even, it could even work if that 200 was 100.
05:35
So what we do is we actually look, okay, well, what happens if i get 50 plus b squared, all right? and 50 minus b squared.
05:46
The reason i'm taking 50 is that two times 50 is 100...