Hi everyone, my name is Ravi Prakash and welcome to Numbers 3. So, we are discussing about prime numbers in Numbers 2. We will continue it here. So, we know that prime numbers in prime numbers, 2 is the, 2 is the only, even, prime. Okay, 2 is the only even prime number. Right, we'll do question also on it.
Very important point. Now, next one. So, see any prime number of the form p square minus 1, right, if p is a prime number, so p square minus 1 is always divisible by 24. It's an important point, right.
p square minus 1 is always divisible by 24. Can easily prove it. Just take p is equal to 6k plus minus 1. Do it whole square, you can prove it. It is always divisible by 24. Okay?
p square minus 1 is always divisible by 24. Now, next one. Next one, digital sum, digital sum of a prime number, of a prime number can never be, can never be 3, 6 or 9. Okay, this is a super important point. Digital sum of a prime number, digital sum of a prime number can never be 3, 6 or 9. Right?
Now, what is digital sum? What is digital sum? Digital sum is adding the digits of a number till we get a single digit, right? That means, let's say 9, 4, 9, 8, 7, 6. What is a digital So, in this mechanism, we first added its 9 plus 4 plus 9 plus 8 plus 7 plus 6, right?
How much it is? 30, 9282, 30, 37, 43. Again, add 43. So, again 43 is written as 4 plus 3. So, what it is? 7. So, 7 is the digital sum for this number 949876, right?
This is the meaning of digital sum, okay? A number, adding the digits of a number till we get a single digit, right? This is a digital sum, okay?
So, Digital sum has got a huge importance in numbers, right? And in mathematics, we'll discuss in algebra also. We'll discuss in application geometry also. In eliminating options, right? Digital sum is very important, right?
So, we'll discuss it. In remainders also, we'll discuss it, right? So, digital sum is very important.
Okay. So, these are the few important points of prime numbers, right? Now, next one.
How to find a number is prime number or not? Okay. This is very important. How to find? This is what we need to have a quick mind, right?
Quick mind. How to find a number is prime or not? Prime or not?
Okay. Now, see here, you take any number, right? You take any number and let's say a square root of n, okay?
n is any number and it is a square root of n, right? So, we check, we check for divisibility, divisibility of, of any number. any number with, we check the divisibility of any number with all the prime numbers, with all the prime numbers before a square root, before a square root of that number.
Okay, so we check for divisibility of any number with all the prime numbers before a square root of that number, right? So, to find a number is prime number or not. For example, if I need to find, suppose 143 is prime number or not? Okay, so 143 is prime or not? Prime or not?
Okay, so we'll check the disability of all the prime numbers before is square root of 143. So, what is square root of 143? So, 12 square is 144, right? So, that means it is square root root of 143 is 11 point something, right?
So, I'll check for all the prime numbers less than 11, less than equal to 11. That is, I'll check with 11, 7, 3, 2, right? These are the prime numbers. So, start checking with 11. So, if you see with 11, so 11 into 13 is 143, right?
It is divisible by 11 times 11, 3 is left, 3 and 3 is 33. So, it is divisible with 143. That means it is not a prime number. It is not a prime number, right? So, I'll tell you the logic why we do so. I'll tell you the logic behind it.
Okay, check for other number. Check if 731. is prime or not. 731 is prime or not, right? So, again, we'll see what is square root of 731. So, 27 square is 729. Approximately, you can take approximately.
So, 27 square is 729. That means, I'll check for all the prime numbers less than 27 because 27 is not a prime number here, right? Here, 11 was a prime number. So, that I started with less than equal to sign because I need to check with 11 also.
Here. 27 is not a prime number. So, what are prime numbers less than 27? 23, 19, 17, 13, 11, like this. I need to check, right?
Till 2, till 2. So, see, if I divide by 23, so if 731 is the number, I divide by 23. So, quickly do it in mind only. 23 into 3 is 69. So, 4 is left, 73. So, 4 is left, 4141 So, not divisible by 23. 19. 19 into 3 is 57. So, 16 is left. 16 into 1 is 61. Now, 19 into 1 is 71, not 1 is 61, right? So, not by 19. 17. 17 forja 68. So, 5 is left.
5 into 1 is 51. 17 is divisible by 51, right? That means it is divisible by 17. It is 17 into 43. So, 731 is not a prime number. Okay.
So, this is how you check a number is prime number or not. Okay. Now, logic, why do we do so? We do so because see, any number, let's say, number is like 32. So, if I write all the factors of 32, right, all the factors of 32, 1, 2, 4, 8, 16 and 32. You can change with any number, right, see for 20 also.
20 also, 1, 2, all the factors of 20, 1, 2, 4, 5, 20, 20. So, if you write all the factors of 32 in ascending order, so 32 can be written as 1 x 32, that is 1 x 32. Then, 2 x 16, 2 x 16. Then, 4 x 8, the 4 x 8, right? Or 20 can be written as 1 x 20, last, first and last, 2 x 10, second and second last, 4 x 5, third and third last, right? So, every number can be written like this.
1 x 20, 2 x 10 and 3 x 3, sorry, not 3 into 4 into 5. See, so it's a property of every number that, and I should, I should, I should take a square root also to explain you further. 36. What is 36? So, 1 into, all the factors of 36, right?
So, 1 into 36, 2 into 18, 3 into 12, and 4 into 9, and then 6 into 6 also, right? 6 into 6 also. Let me write here, 6 into 6. Okay, so what factors of 36 are?
1, 2, 3, 4. 1, 2, 3, 4, 6, 9, 12, 18, 36. See, so it's the property of every number that once you write a number as the product of two factors, right? So, look here, like a square, all these three factors, right? If there are six factors, three factors will lie before the square root, three factors will lie after the square root. Like a square root of 32 is around 5 point, let's say 5, because 5 point, right?
So, see, if a square root is 5.5, So, 3 factors are less than 5.5, 3 factors are more than 5.5. Square root of 20 is around 4.5, 4 square root of 16 is around 5 square root of 25, so around 4.5, right? So, see 4.5.
So, 3 factors less than 4.5, 3 factors more than 4.5. Square root of 36 is what? 6, 6, right?
6 is square root. So, 4 factors are less than 6, 4 factors are more than 6, right? basically means that if if for 36, if 36 is divisible by 2, then I don't need to check for 18. If 36 is divisible by 3, I don't need to check by 12. Why? Because there's a 3 and 12 are corresponding terms, 2 and 18 are corresponding terms, 4 and 9 are corresponding terms, right?
Corresponding terms. So, if 4 is divisible by 36 and 4 is a number before a square root, its corresponding term will exist after square root also, that is 9. So, if I check by 4, I don't need to check by 9. If I check by 3, I don't need to check by 12, right? That means, all the if at all, a number's factors exist. as many factors before the square root are there, same number of factors after the square root are there, right. So, if we check for all the factors before the square root, we do not need to check for all the factors after square root.
That is why for any number, if I check the divisibility or if it is prime number or not, right, for prime number or not, what we are checking? We are checking the divisibility only. That number is divisible by all the numbers before its square root, all the prime numbers before the square root, right, all the prime numbers because any composite number or any other number can be broken in terms of prime number, right? Like 8. 8 is 2q. So, if a number is not divisible by 2, right?
If a number is not divisible by 2, it won't be divisible by 8, never, right? Suppose 6 is 2 into 3. So, if a number is not divisible by 6, not divisible by 2, it can never be divisible by 6, right? So, that's why we're checking for only prime numbers and that to be for the square root, right? This is the whole logic in the prime.
numbers. Right. So, many times the question comes right, following numbers are prime or not.
So, how to check numbers is prime or not? How to check numbers is number is prime or not? We have got the four different way to check it. Right. First way is what?
First way, I can write digital sum or I should write in fact, first way is all prime numbers are of the form 6k plus minus 1. Second way I can write digital sum of a prime number can never be Digital sum can never be, never be 3, 6 or 9. Third way I can write, p square minus 1 is always divisible by 24. That means p square minus 1 is always a multiple of 24, right? Fourth one I can write, before, before square root of n. Check before square root of n, right?
There are four different ways to check a number is prime number or, okay? I hope it is clear, right? fine okay so if i want to check right if i want to check one or okay okay i will do questions right we'll do we'll do questions here okay let me do few questions then we can we'll be sure of this concept okay now question okay if you are doing questions let me explain also what are composite numbers so what are composite numbers so composite numbers are all the numbers all the natural numbers, numbers having More than, having, more than or equal to 3 factors.
Right, all the natural numbers having more than or equal to 3 factors are component numbers. Right, for example, 9. So, 9 is a component number having 3 factors, 1, 3, 9. Okay, for example, 15. It's a component number having 3 factors. 1, 3, 1, 3, 5, 15. That means 4 factor, right? That is more than 3 factor. So, more than or equal to 3 factors are what?
Composite numbers hundred it will having more than three factors, right? So these are all composite numbers Okay, having three factors or more than three factors so you can write here that one one is Neither prime one is neither prime nor Composite okay one is neither prime nor composite one is neither prime nor composite because one is having only one factor so prime numbers are having two factors and come Composite numbers are having greater than equal to 3 factors, right? So, 1 doesn't lie in any of these 2 categories. So, 1 is neither prime nor composite.
Okay, now we can do few questions here. Okay, let's discuss. Is 3 raised to 193 plus 5 prime or composite number?
Okay, the question is, is 3 raised to 193 plus 5 prime or composite numbers? Can we solve it? See, 3 raised to 193 is an odd number.
Why odd number? Because 3 is an odd number and odd number, any power will always be odd number, right? Like 3, 3 square is 9, 3 cube 27, 3 raised to 4, 81, right?
That means odd number, any odd number is multiplied, right? 2 times, 3 times, 4 times, 5 times, it remains odd, it remains odd, right? That means odd number, any power, odd number to any power will always be Odd will always be odd, right?
That means 3 raised to 193 is an odd number, right? So, 3 raised to 193 is an odd number and this plus 5, 5 is also an odd number and sum of two odd numbers is always even number, right? Like take an example, 3 plus 5 is 8, 5 plus 9 is 14. Sum of two odd numbers is always even number right?
That means this number is an even number okay and an even number can never be prime except 2, right, except 2. Except 2, no other even number is prime. Therefore, this number is what? This number is composite. It is not a prime number.
Okay. So, except 2, no other even number is prime number. All prime numbers are odd numbers. Right. Only 2 is the even prime.
So, 3 raised to 193 plus 5 is an even number, which is not possible to be a prime number because prime numbers are always odd numbers except 2. Okay. So, this is the question. Correct?
Now, next one is is you see 1,000, 1 and 4 times 0, right? 1 and 5 times 0, you put 1,000, 1, you can write as 1,00,000, 1. Is 1,00,000, 1 prime or composite? Now, we'll discuss few concepts here, right?
So, is 1,00,000, 1 prime or composite, right? So, see 1,00,000, 1 can be broken in the form of 10,00,000, right? There are 3 digits here. So, 10,00,000. So, it is like 10 raised to 6 plus 1. So, 1 lakh 1 can be written in the form of 10 raised to 6 plus 1. Okay, 10 raised to 6 plus 1. Right?
10 raised to means 6 means 6 zeros. Right? Am I correct, I think? Sorry, I have written here 4 zeros only.
Here 5 zeros here. It will be 5 zeros here. Okay.
So, 5 zeros. Huh. So, is 1 lakh 1 prime or composite? It is not 1 lakh 1. It is not 10 lakh 1. Is 10 lakh 1 prime or composite? Right?
So, 10 lakh is? 10 raised to 6 plus 1. Now, see 10 raised to 6 can be written in the form of, now see here, 10 raised to 6 is what? 10 is square cube and it is 1, 1 is 1 cube.
Okay, 1 is what? 1 is 1 cube. Okay, so 10 a cube plus b cube, right, this concept.
What is a cube plus b cube now? See, so a cube plus b cube is always what? a plus b into a is square minus ab plus b square.
Right, so this is an identity a cube plus b cube. This is a plus b into a square minus a plus b square. Right, so 10 square whole cube plus 1 cube is always going to be what?
It will be like 10 square plus 1 into something. Right, that means it is 101 into something. It is something that we don't need.
Right, simply we need to check it is prime or not. So, it is 101 into something and 101 into something is always right that means it's a multiple of 101 so it can't be a prime number it can't be a prime number it can't be a prime number right so it is divisible by this number is this number is divisible by 101 we can say right one more thing i'll tell you it is also from a raised to n plus b raised to n. Okay. So see here.
See a cube plus b cube is equal to a plus b into a square plus a b plus b square a cube minus b cube is equal to a minus b into a square sorry sorry minus here a square minus a b plus b square a cube minus b cube is a minus b into a square plus a b plus b square right a square minus b square is a minus b into a plus b correct a square P plus B square cannot be written in factor form, cannot be factorized. Factorized means cannot be written as a product of terms, right? It can't be factorized, a square plus b square, right?
a4 minus b4. Again, you see it is a minus b into something. So, we can conclude from here that, see, we can conclude from here that whenever it is of the form a raised to n plus b raised to n, number is of the form a raised to n plus b raised to n, right? So, when n is odd, right, When n is odd, first case I'll take when n is odd. So, when n is odd, in that case, right, in that case, like a cube plus b cube, okay, it is always divisible by, always divisible by a plus b, okay, always divisible by a plus b.
n is even, n is even, like a square plus b square. Yeah, something, anything cannot be concluded. If a is, if it is written a n plus b n, right, aN plus bN, it always cannot be concluded, right?
Anything cannot be concluded. Anything cannot be concluded, right? Anything cannot be concluded. Now, second part, aN minus bN, okay?
And now, when n is odd, when n is odd, like a cube minus b cube, it is always divisible by a So, always divisible by a minus b. And when n is even, n is even, right? Like a square minus b square.
Like a square minus b square. So, see I've written then few mistakes, right? Not few mistakes actually, but problem in pen here. So, it is a minus b into a plus b. You make it a minus b into a plus b.
a square minus b square is a minus b into a plus b, right? Sometimes it is like missing. Okay, no issue.
I'll correct it, right? you will not be taught anything wrong don't worry okay So, a n plus b n is okay. So, when n is like now n is even. So, in this case now n is even. Okay, so when n is even.
So, a n plus b n when always n is odd, right. So, a cube minus b cube it is already divisible by a minus b. Now, when a is n is even like a square minus b square. Like a a square minus b square. Okay, so when n is even, it is already divisible by a plus b also.
Okay. always divisible by a plus b also, right? Like in this case, a4 minus a4 minus b4, okay?
So, a4 minus b4 will always be a minus b into a plus b into something, right? Will always be a minus b into a plus b into something, right? I hope it is clear here.
Okay, now let's move to next one Let's move to, yeah, okay, see here. So, if I write a cube minus b cube, it is a minus b into a square plus ab plus b square, right? Okay, if I write a cube plus b cube, it is a plus b into a square minus ab plus b square, right?
If I write a square minus b square, it is a plus b. into a minus b. If I write a square plus b square, right, it cannot be factorized.
It cannot be factorized. Means, I can't write in terms of any two products like this or three products, okay. Then a4 minus b4 is also a minus b into a plus b into something.
It goes like this, right. So, what I am trying to explain you here is, see, if a number right, if a number is of the form a raised to n plus b raised to n, okay. And if n is odd, okay, first case I'll take n is odd, just compared to the most basic formula, right.
So, if n is odd, like here, if n is odd, so a cube plus b cube, right. So, if n is odd, a cube plus b cube is always divisible by a plus b. So, always any number of the form a raised to n plus b raised to n, If n is odd, always divisible by a plus b. Okay, second, if n is, if n is even, right, if n is even.
So, if n is even, like a square plus b square, no definite conclusion, right, no definite conclusion you can make here, right, for n to be, when n is even, right. Now, say second one. Now, of the form a raise to n minus b raised to n and we'll discuss again first case when n is odd. Now, n is odd like a cube plus b cube it is always sorry like a cube minus b cube it is always divisible by a minus b.
Okay, so always always any number of the form a raised to n minus a raised to n minus b raised to n when n is odd is always divisible by a minus b. Second case when n is even you see when n is even also right like a square minus b square like a 4 minus b 4 always double by a plus b and a minus b right so always divisible by a plus b and a minus b both both right so very important points we have discussed here okay so let's do some questions here again is is 2 raised to 2 raised to 3007 plus 1, prime or composite, prime or composite. Okay, so how to find it?
So, we have just discussed this. We have just discussed this. Any number of the form, right, any number of the form, a n plus b n is always divisible by, is always divisible by a plus b.
Always, always divisible by a plus b. Sorry, a plus b, right. So, 2 raised to 3007 plus, let's say, I can write like 1 raised to 3007. So, a plus b.
So, a plus b, always divisible by, so it is always divisible by 3, right? So, very important, this concept, right? Always divisible by 3. Or for all the numbers close to, close to some 10 powers, you can easily do it, right? Close to some 10 powers, you can easily do it, right?
If I write is is 973 prime or composite. Right, it's a small number. What you can do is directly apply that checking before root n.
That also you can do. But second is also useful as it is close to power of 10. So, I can write 973 as 1000 minus 27. That is 10 cube minus 3 cube. So, a cube minus b cube is always divisible by a minus b. a raised to n minus b raised to n is always divisible by a minus b if n is odd, right?
This is always divisible, right? This is always divisible by 7, okay? Always divisible by 7, right?
This is very important concepts we are discussing here about prime numbers, good questions we are discussing. How to find a number is prime number or not? Okay, so we'll do in the further video okay thank you