Question

using the prime factorization method find which of the following number are perfect squares 189, 225, 2048, 343, 2916, 11025 , 3549​

Answer 1

Given:

1)189 2)225 3)2048 4)343 5)2916 6)11025 6)3549

To Find:

Using prime factorization method find which of the above numbers are perfect square.

Solution:

If the total number of factors of a given number are odd, the number is a perfect square.

Let N be a number

Using prime factorization method, calculate the total number of factors of the given number

$N= a^{p} b^{q} c^{r}$       ;

where a ,b and c are prime factors of number N and p,q and r are the power raised of the prime factors respectively.

Total number of factors = n =$(p+1)(q+1)(r+1)$

1)189

189 = 3×3×3×7= $3^{3} 7^{1}$     ;

n = (3+1)(1+1) = 8 = not a odd number

Hence 189 is not a perfect square.

2)225

225= $3^{2} 5^{2}$       ;

n = (2+1)(2+1) = 9 = odd number

hence 225 is a perfect square = $15^{2}$

3) 2048

2048 = $2^{11}$    ;

n = 11+1 = 12= not a odd number  

hence 2048 is not a perfect square.

4)343

343 = $7^{3}$     ;

n = 3+1 = 4

not a odd number hence 343 is not a perfect square.

5)2916

2916=$2^{2} 3^{6}$      ;

n=$(2+1)(6+1) = 21$

odd number , Hence 2916 is a perfect square

6)11025

11025= $5^{2} 7^{2} 3^{2}$     ;

n =$(2+1)(2+1)(2+1) = 27$ which is a odd number

therefore 11025 is a perfect square = $105^{2}$

7)3549

3549= $7^{1} 13^{2} 3^{1}$   ;

n = (1+1)(2+1)(3+1)=24 which is a even number hence 3549 is not a perfect square.

1)189 - not a perfect square

2)225- perfect square

3)2048- not a perfect square

4)343- not a perfect square

5)2916-perfect square

6)11025-perfect square

7)3549- not a perfect square

 

Answer 2

Answer:

189 is not a perfect square because, it does not pair with each other