Use Euclid's division lemma to show that the cube of any positive integer is of the form 9m ,9m+1,or9m+8 ? 5. Use Euclid's division lemma to show that the cube of any positive integer is of the form9m, 9m+1 or 9m+8.1.3 The Fundamental Theorem of ArithmeticIn your earlier classes, you have seen that any natural number can be written as aproduct of its prime factors. For instance, 2 2, 4 2 x 2, 253 = 11 x 23, and so on.Now, let us try and look at natural numbers from the other direction. That is, can anynatural number be obtained by multiplying prime numbers? Let us see.Take any collection of prime numbers, say 2, 3, 7, 11 and 23. If we multiplysome or all of these numbers, allowing them to repeat as many times as we wish,we can produce a large collection of posiLet us list a few:integers (In fact, infinitely many).7x 11 x 23 17713 x7x11 x 23 = 53132x 3x7x 11 x 23 = 106262 x 3 x 7 823222 x 3 x 7x 11 x 23 = 21252and so on.Now, let us suppose your collection of primes includes all the possible primes.What is your guess about the size of this collection? Does it contain only a finitenumber of integers, or infinitely many? Infact, there are infinitely many primes. So, ifwe ombine all these primes in all possible ways, we will get an infinite collection ofSHOT ON OPPO A SHOT ON OPPO5. Use Euclid's division lemma to show that the cube of any positive integer is of the form9m, 9m+ 1 or 9m+8.3 The Fundamental Theorem of Arithmetica your earlier classes, you have seen that any natural number can be written as aroduct of its prime factors. For instance, 2 = 2, 4 = 2 x 2, 253 = 11 × 23, and so on.Fow, let us try and look at natural numbers from the other direction. That is, can anyatural number be obtained by multiplying prime numbers? Let us see.Take any collection of prime numbers, say 2, 3, 7, 11 and 23. If we multiplyme or all of these numbers, allowing them to repeat as many times as we wish,e can produce a large collection of positive integers (Int us list a few :infinitely many).7x 11 x 23 = 17713 x 7x 11 x 23 = 53132x 3x 7x 11 x 23 = 106262x 3 x 7x 11 x 23 = 212522 x 3 x 73 8232SO on.Now, let us suppose your collection of primes includes all the possible primes.is your guess about the size of this collection? Does it contain only a finiteer of integers, or infinitely many? Infact, there are infinitely many primes. So, ifmbine all these primes in all possible ways, we will get an infinite collection ofers, all the primes and all possible products of primes,on is

Name: Use Euclid's division lemma to show that the cube of any positive inte
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Description: Use Euclid's division lemma to show that the cube of any positive integer is of the form 9m ,9m+1,or9m+8 ? 5. Use Euclid's division lemma to show that the cube of any positive integer is of the form9m, 9m+1 or 9m+8.1.3 The Fundamental Theorem of Arit