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) 25a^{2}-35a+12 (II) 35y^{2}+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

^{2}-35a+12 (II) 35y

^{2}+13y-12

(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.

##
Find the HCF and LCM of 90 and 144 by the method of prime factorization.

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

^{n}can be end with the digit 0 for any natural number n.

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

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

^{n}cannot end with the digit zero, for any natural number n.

7^{n} =(1x 7)^{n} = 1^{n} x 7^{n}

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

7^{n} 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 = 3^{2} x 13

LCM = 13 x 5 x 3^{2} = 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 3^{3} x 7

180 = 2^{2} x 3^{2} x 5

420 = 2^{2} 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 = 2^{3} x 3^{2} x 5 x 7

10530 = 2 x 3^{4} x 5 x 13

LCM = 2^{3} x 3^{4} x 5 x 7 x 13

= 294840

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

^{n},n being number, end with the digits 5 ? Given reasons

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

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

6^{n} = (2 x 3)^{n} = 2^{n} x 3^{n}

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

6^{n} can never end with 0.

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

1936 = 4^{2} x 11^{2} = 44^{2}

##
Find the perfect cube by the prime factorization method for the following?

a. 729

b. 5832

9x9x9 = 729

##
Divide :

(i)

(ii)

(i) For taking 3 common

=

For correct answer = xy

(ii) For factorization of

For correct answer = y+2

##
a. What must be subtracted from 4562 to get a perfect square? Also, find the square root of this perfect square.

b. Find the square root by factorization method 14400.

33