How many factors of 10800 are perfect squares?

Question

A. 4
B. 6
C. 8
D. 10
E. 12

niks18 · Accepted Answer

10800=1*2^4*3^3*5^2

For a factor to be a square it needs to have an even number of powers of each of the prime factors, in this case for  2 ,  3  &  5

so for the sake of explanation, let  10800=2^a*3^b*5^c

Now  a  can take values  0 ,  2  &  4  i.e  3  values

b  can take values  0  &  2  i.e.  2  values

and  c  can take values  0  &  2  i.e.  2  values

Hence total number of perfect squares  = 3*2*2=12

Again just to explain why we need to multiply here -
All the square factors occur when we take combinations of exponents from the three sets - {0,2,4}, {0,2} & {0,2}. hence the rule of multiplication is applied here for counting

Option  E

chetan2u · Answer

hi..

Factors of  10800=1*2^5*3^3*5^2 
all prime factors have atleast power of 2

so ways..
 1) single digits.. 
1,2,3,4,5.... so 5 of them
 2) product of two prime factors.. 
2*3
2*5
3*5
3*4
4*5
so 5 ways
 3) product of 3 prime factors 
2*3*5
3*4*5
so 2 ways

total = 5+5+2=12 ways
E

mahu101 · Answer

E. 12

10800 =  3^3 * 2^4 * 5^2

For a factor to be perfect square it needs to have even powers of 3, 2, 5.
Hence we count the number of even exponents of 3, 2, 5 and multiply them (combination activity)

The exponent count would be 
0 and 2 for 3 - (total of 2)
0, 2, and 4 for 2 (total of 3)
0 and 2 for 5 (total of 2)

Total number of perfect square factor is 2 x 3 x 2 = 12
Why zero is included? because 1 (for example  2^0 * 5^0  = 1 x 1) is also a factor and it is a square.

Take an easy number 36 - How many factors of 36 are perfect squares?
36 =  3^2 * 2^2 
Even exponents - 0, 2 for 2 and 0, 2 for 3
Total exponent count is two each for 2 and 3
Hence perfect square factors are 2 x 2 = 4

Long method
Factors of 36 = 1, 2, 3, 4, 6, 8, 12, 36
Perfect sq factors are 1, 4, 9 and 36 - Hence 4 factors are perfect squares.

niks18 · Answer

Hi  chetan2u

You have got the right answer but I am not able to understand your approach. can you explain it in more detail.
Also I don't think we can use summation for number of ways here.

amanvermagmat · Answer

Another way:

10800 = 2^4 * 5^2 * 3^3

Perfect squares are those where all powers are even (multiples of 2). First step thus is to take out something common and leave in parenthesis everything that is a perfect square.

So 10800 = 3 * (2^4 * 5^2 * 3^2). Now whatever is inside parenthesis is a perfect square. Lets see what is it the square of ?

We can re-write this as: 3 * (2^2 * 5 * 3)^2. 
Note that the expression after 3 is whole square of (2^2 * 5 * 3). Now how many factors does 2^2 * 5 * 3 have? 3*2*2 = 12.

That's our answer. (basically whatever factor you take out of 2^2 * 5 * 3 will give you a new perfect square because its already raised to a power of 2)

This given number 10800 has 12 such factors which are perfect squares. Hence  E answer

chetan2u · Answer

Hi...

the first step has been to factorize 10800, which is 2^5*3^3*5^2...
here all are atleast to POWER of 2 and 2^5 also includes 4^2

what is left is to find different combinations or choosing 1 or more out of prime factors - 1,2,3,2^2,5
single - 1,2,3,4,5 THAT is it contains  1^2,2^2,3^2,4^2,5^2 
two at a time - 2*3,2*5,3*5,3*4,4*5, THAT is it contains  (2*3)^2, (2*5)^2....... 
three at a time - 2*3*5, 3*4*5 THAT is it contains  (2*3*5)^2  and  (5*4*3)^2

total 12

hellosanthosh2k2 · Answer

Prime factorization:  10800  =  2^4 * 5^2 * 3^3 
 =  4^2 * 25^1 * 9^1 * 3

To get factors which are perfect squares : we need to consider factors from :  4^2 * 25^1 * 9^1 
 
set A: {  4^0, 4^1, 4^2  } = 3 (total) 
set B: {  9^0, 9^1  } = 2 (total)
set C: {  25^0, 25^1  } = 2 (total)

Total combinations : 3 * 2 * 2 (picking anyone from each set) = 12 => (E)

Adit_ · Answer

Small correction 10800 = 2^4*3^3*5^2 not 2^5..

How many factors of 10800 are perfect squares?

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How many factors of 10800 are perfect squares?

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