SOLUTION: What is the number of factors of 222264 that are perfect squares?

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Question 1200580: What is the number of factors of 222264 that are perfect squares?
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52770)   (Show Source): You can put this solution on YOUR website!
.
What is the number of factors of 222264 that are perfect squares?
~~~~~~~~~~~~~~~~~~~~~ 

Prime decomposition of the number  222264  is

    222264 = .


The factors that are perfect squares are

    1, , , , ,

       , ,

       ,

       , ,

       , .


In all, there are 12 factors that are perfect squares.     ANSWER

Solved.



Answer by greenestamps(13198)   (Show Source): You can put this solution on YOUR website!


By whatever means you want, determine that the prime factorization of 222264 is

(2^3)(3^4)(7^3)

So any factor of 222264 is of the form

(2^a)(3^b)(7^3)

where
0<=a<=3
0<=b<=4
0<=c<=3

If the factor is a perfect square, then a, b, and c all have to be even. So

a can be 0 or 2 (2 choices)
b can be 0, 2, or 4 (3 choices)
c can be 0 or 2 (2 choices)

By the fundamental counting principle, the number of factors of 222264 that are perfect squares is 2*3*2 = 12.

ANSWER: 12
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