i did this by brute force because i didn't know any formula that would allow me to find it directly.
there may be one but i don't know what it is.
the brute force method used excel.
i simply multiplied 3456 by numbers starting from 1 and going up and then checked to see if i got a perfect square or not.
the answer is 6
3456 * 6 is a perfect square and multiplying by 6 is the smallest integer i could multiply 3456 by before getting a perfect square as the result.
here's the results of the brute force method.
3456 1 58.78775383
3456 2 83.13843876
3456 3 101.8233765
3456 4 117.5755077
3456 5 131.4534138
3456 6 144 this is a perfect square
3456 7 155.5377768
3456 8 166.2768775
3456 9 176.3632615
3456 10 185.9032006
3456 11 194.9769217
3456 12 203.646753
3456 13 211.9622608
3456 14 219.9636334
3456 15 227.6839915
3456 16 235.1510153
3456 17 242.3881185
3456 18 249.4153163
3456 19 256.249878
3456 20 262.9068276
3456 21 269.3993318
3456 22 275.739007
3456 23 281.936163
3456 24 288 this is a perfect square
3456 25 293.9387691
if there is a rule, i don't know it, as i said before.
it's interesting that the next perfect square is multiplying by a multiple of 6, specifically 4 * 6.
note that 6 by itself is not a perfect square even though 4 by itself is a perfect square.
no real relationship there that i can figure out.
if you're looking for a formula to determine the smallest multiple to get a perfect square i don't have it, but you do have your answer is that's all you wanted.
i did a search of the web and i did find a way to determine it.
the link i found is here:
http://answers.yahoo.com/question/index?qid=20090616105542AAKgSOq
basically, you break the number down into factors and each factor has to be a perfect square.
if there's a number that's not a perfect square, then that number is the number you have to multiply your original number by in order to get it to be a perfect square.
applying that concept to your number, we get the following:
3456 / 144 = 24 which means that:
3456 = 144 * 24
144 is a perfect square, but 24 is not.
we can, however break down 24 into a multiple of perfect squares.
24 = 2*12 = 3*8 = 4*6
4 is a perfect square, so we get:
3456 = 144*4*6
since 144*4 is equal to 576 and 576 is a perfect square (24*24), we can reduce this to:
3456 = 576*6
6 is not a perfect square, but if we multiply it by 6 we will get a perfect square, then that's the smallest we can get.
this is because:
6 is not a perfect square but it can be broken down into:
6 = 2*3 = 1*6 neither of which is a perfect square.
that makes 6 the smallest multiple of 3456 to get a perfect square.
brute force method confirms that:
6 1 2.449489743
6 2 3.464101615
6 3 4.242640687
6 4 4.898979486
6 5 5.477225575
6 6 6 this is a perfrect square
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