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We know LCM of three numbers can be calculated by writing the numbers in their prime factorization.

\[

6 = 2 \times 3 \\

9 = 3 \times 3 \\

15 = 3 \times 5 \\

\]

So to find the LCM we will take the highest number of times whichever prime is repeated and multiply with other primes.

We have 2 only one time, 3 is multiplied two times so we take \[3 \times 3\] and 5 is also there one time.

SO, LCM is \[2 \times 3 \times 3 \times 5 = 90\]

Now we know prime factorization of \[90 = 2 \times 3 \times 3 \times 5\]

To find the smallest number that is a square and is divisible by 90 we will form a number by multiplying factors to the prime factorization of 90.

So, to make \[2 \times 3 \times 3 \times 5\] a square we need each number to be multiplied to itself. i.e. each prime factor should be in square.

We can write the prime factors in the form of their powers as

\[ \Rightarrow 2 \times 3 \times 3 \times 5 = {2^1} \times {3^2} \times {5^1}\]

Now we can see 3 has a square power but 2 and 5 do not have square power. So we multiply with 2 and 5

\[

= {2^1} \times 2 \times {3^2} \times {5^1} \times 5 \\

= {2^2} \times {3^2} \times {5^2} \\

= 4 \times 9 \times 25 \\

= 900 \\

\]

So, the smallest square number that can be divided by 6, 9 and 15 is 900.