How many factors of 1764 are perfect squares ? A. 4 B. 7 C.

Question

How many factors of 1764 are perfect squares ?
A. 4
B. 7
C. 8
D. 9
E. 10

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nick1816 · Accepted Answer

1764=2^2*3^2*7^2

Hence any factor of 1764 can be written in the form of  2^a*3^b*7^c 
where 0≤a≤2, 0≤b≤2 and 0≤c≤2

If the factor is perfect square, then a, b and c must be even.

a can take 2 values (0 or 2)
b can take 2 values (0 or 2)
c can take 2 values (0 or 2)

Number of factors of 1764 are perfect squares= 2*2*2=8

pushpitkc · Answer

Prime factorizing 1764, we get  2^2*3^2*7^2

The factors of 1764 that are perfect squares are 1,4,9,36,49,196,441 and 1764(Option C)

CrackverbalGMAT · Answer

This is a fairly straightforward question on the topic of Factors & Multiples. So, the best approach is to prime factorise the given number and then use the formula.

If N is a composite number which can be written as N = \(a^p * b^q * c^r *\)…………, then the number of factors of N = (p+1) * (q+1) * (r+1)*……

1764 = \(2^2 * 3^2 * 7^2\).

From this we gather that 1764 itself is a perfect square. Therefore, all the prime factors of 1764 have even powers. Also, the total number of factors = (2+1) * (2+1) * (2+1) = 3*3*3 = 27.

Now, how do we find out how many of these are perfect squares? Let’s look at the diagram below:

Attachment:

25th June 2019 - Reply 2.JPG [ 35 KiB | Viewed 5798 times ]

Under each prime factor, you see that there are 2 options listed out. Since there are 3 prime factors, we can form a total of 8 combinations.

For example, \(2^0 * 3^0*7^0\) = 1 is a factor of 1764 and is a perfect square. Similarly, \(2^0 * 3^2 * 7^0\) is also a factor of 1764 and a perfect square.
This way, we can have 8 such products which are all factors of 1764 and are perfect squares. Therefore, the correct answer has to be option C.

Hope this helps!

jfranciscocuencag · Answer

The easiest way I found but a little time consuming was just take the LCM of the number:

1764=2^2*3^2*7^2

Then as all the factors (2,3 and 7) are perfect squares, we play with the factors:

2^2 --- 1 
 3^2 --- 2 
 7^2 --- 3 
 2^2*3^2*7^2  All together (2*3*7)^2 --- 4
 2^2*3^2 --- 5 
 2^2*7^2 --- 6 
 3^2*7^2 --- 7

And finally do not forget 1 --- 8 factors in total.

Kinshook · Answer

Asked: How many factors of 1764 are perfect squares ?

1764=2^2*3^2*7^2

Perfect squares are made when power of 2,3 and 7 are either 0 or 2.

Total ways = 2*2*2=8

IMO C

Basshead · Answer

Can we simply multiply the even powers? Or is it necessary to list the powers in 0?

How many factors of 1764 are perfect squares ? A. 4 B. 7 C.

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How many factors of 1764 are perfect squares ? A. 4 B. 7 C.

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