To find the least number that is a perfect square and divisible by 16, 18, and 45, we need to find the least common multiple (LCM) of these three numbers and then find the smallest perfect square that is divisible by the LCM.
First, let's find the LCM of 16, 18, and 45.
Prime factorization of 16:
16 = 2^4
Prime factorization of 18:
18 = 2 * 3^2
Prime factorization of 45:
45 = 3^2 * 5
To find the LCM, we take the highest powers of all prime factors involved:
LCM = 2^4 * 3^2 * 5 = 720
Now, we need to find the smallest perfect square that is divisible by 720. We can do this by taking the square of the prime factorization with each exponent divided by 2:
720 = 2^4 * 3^2 * 5
Taking the square root of each factor:
√(2^4 * 3^2 * 5) = 2^2 * 3 * √5 = 12√5
So the smallest perfect square divisible by 16, 18, and 45 is (12√5)^2.
Calculating the square:
(12√5)^2 = (12^2) * (√5)^2 = 144 * 5 = 720.
Therefore, the least number that is a perfect square and divisible by 16, 18, and 45 is 720.