SOLUTION: By what number should the number 19404 be divided to get a perfect square?also find the number whose square is the new number.

Question 1118674: By what number should the number 19404 be divided to get a perfect square?also find the number whose square is the new number.
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

Find perfect square factors of the given number and divide by those perfect squares; continue until what is left is NOT a perfect square. That will be the answer to the first question.

Note you only need to check for squares of prime numbers; if you have already found that the remaining number is not divisible by 2^2=4, then it won't be divisible by the square of any other even number.

The task is easy for me, because mental arithmetic is easy for me. For most people, a general strategy would be simply to check for divisibility by squares of the smallest prime numbers.

Inspection shows that 19404 is divisible by 4, because the last two digits are divisible by 4: 19404 = 4*4851.

The remaining number is not divisible by 2^2=4, so next check 3^2=9.

Inspection next shows the number is divisible by 9, because the sum of the digits is divisible by 9: 4851 = 9*539.

Those first two steps were easy if you know some basic divisibility rules. But it gets only a little bit harder from here if you follow the basic process of checking for divisibility by squares of prime numbers.

Clearly the number remaining is not divisible by 5^2=25, so try 7^2=49: 539 = 49*11.

So the original number is

19404 = 4*9*49*11

Now we can answer the first question: Dividing 19404 by 11 yields a perfect square.

From there the second question is easy. Dividing 19404 by 11 yields

4*9*49 = (2^2)(3^2)(7^2)

So the number whose square is the new number is 2*3*7 = 42.
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