What can you say about the prime factorization of the denominator of 27.142857

What can you say about the prime factorization of the denominator of 27.142857

## Ram has two rectangles having their areas as given below: (1) 25a2-35a+12                   (II) 35y2+13y-12 (i) Given possible expressions for the length and breadth of each rectangle. (ii) Which mathematical concept is used in this problem? (iii) Which value is depicted in this problem

(i) Possible length and breadth of the rectangle are the factors of its given area.

Area    =

So possible length and breadth are (5a -3) and (5a - 4) units, respectively.

(ii)  Area  =

So, possible length and breadth are (7y-3) and (5y+4) units respectively.

(ii) Factorization of polynomials

(iii) Expression of one's desires and news is very necessary.

## Check whether 4n can be  end with the digit  0  for any natural number n.

If the number 4n, for any n, were to end with the digit zero, then it would be divisible by 5.That is, the prime factorization of 4n would contain the prime  5. This is not possible  because 4n = (2)2n ; so the only prime in the factorization of 4n is2. So, the uniqueness of the Fundamental Theorem of arithmetic guarantees that there are no other primes in the factorization of 4n. So there is no natural number  n for which 4n ends with the digit zero

## Show that 7n cannot end with the digit zero, for any natural number n.

7n =(1x 7)n = 1n x 7n

So the only prime in the factorization of  7n is  7, not 2 or  5.

7n cannot end will the digit zero.

## The HCF of 65 and 117 is expressible in the form 65m-117. Find the value of m. Also find the LCM of 65 and 117 using prime factorization method.

We have,

117  = 65 x 1 +52

65 = 52 x 1 +13

and 52 = 13 x 4 +0

Hence. HCF = 13

65m -117 = 13

65m = 117 +13 = 130

m = 130/65 = 2

Now, 65 = 13 x  5

117 = 32 x 13

LCM  = 13 x 5 x 32 = 585

## Find HCF of 378, 180  and 420 by prime factorization method. Is HCF x LCM  of three  numbers equal to the product of the three numbers ?

Prime factors  of :

378 = 2 x 33 x 7

180 = 22 x 32 x 5

420 = 22 x 3 x 7 x 5

HCF = 2 x 3

= 6

No  of because HCF x LCM     product of three numbers.

## State fundamental theorem of arithmetic. Find LCM numbers 2520 and  10530 by prime factorization method

Fundamental theorem of arithmetic : Every  composite number  can be expressed as the product of powers of primes and this factorization in unique.

2520 = 23 x 32 x 5 x 7

10530 = 2 x 34 x 5 x 13

LCM  = 23 x 34 x 5 x 7 x 13

= 294840

## Can the number 6n,n  being number, end with the digits 5 ?  Given reasons

If 6n ends with 0, then it must have 5 as the factor

But we know that only prime factors 6n are  2 and 3.

6n = (2 x 3)n  = 2n x 3n

From the fundamental theorem of arithmetic, we know that prime factorization of every composite numbers is unique.

6n can never end with 0.

## Find the square root of 1936 by prime factorization metho?

1936 = 42 x 112 = 442

9x9x9 = 729

## Divide :  (i)   (ii)

(i) For taking 3 common