really really hard math problem, i need help?

Question

what is the smallest perfect square that is greater than 20 000 and is divisible by 392????????

Answer 1

the number is 28,224.
since 392=2^3 * 7^2,

then a perfect square divisible by 392 needs to have 3 twos and 2 sevens.

so the root of this perfect square should have at least 2 twos and one seven.

but (2^2 * 7)^2= (28)^2=784, which is less than 20,000

so the next one (2^3 * 3 * 7)^2= (168)^2= 28,224

Answer 2

I will assume that divisible by 392 means that the answer is an integer. So the problem is

N x 392 > 20000 where N is an integer

M x M = N x 392 where M is an integer.

We know that N > 20000/392 which is 51.02048...

The factors of 392 are 2 x 2 x 2 x 7 x 7 (an odd number of 2s)

So we know that M must have 2 factors of 2 in it.

So M must have factors 2 x 2 x 7 x ?

So we now want the smallest square that is > 25.5. The next square is 36.

Therefore N = 72 and M = 2 x 2 x 7 x 6 = 168

and M x M = 28,224

Answer 3

This is actually not that hard. First, look at the information that is given to you. What does it mean for a number to be divisible by 392?

392 = 2*196 = 2*14*14 = 2*2*2*7*7

So if the number we are looking for is x, then x = 2*2*2*7*7*y for some y. Now, since x is also a perfect square, there must be an even number of 2's in its factorization. Thus

x = 2*2*2*2*7*7*z for some z (which is, itself, a perfect square).

Now we use that x > 20,000, so 2*2*2*2*7*7*z > 20,000, so solving this we get

z > 25.51 and z must be a perfect square. The smallest such z is 36, so the answer you are looking for is:

x = 16*49*z = 16*49*36 = 28224

Answer 4

28224

How I arrived this is as follows
392 = 7 x 7 x 2 x 2 x 2
.........____....____
multiplying with 2 gives 784, which is a perfect square of 28.

Dividing 20,000 by 784 = 25.51, the next perfect square is 36.
784 x 36 = 28224, which is a perfect square of 168

Answer 5

168 squared, or 28224, which is divisible by 392

Addendum: It's rare to see everyone get the exact right answer. Kudos, everyone. Now, let's see who gets picked best. Short stick, anyone?