C=216k, find the least value of k that will make C a perfect square
We can start by factoring 216:
216 = 2^3 × 3^3
To make C a perfect square, we need to have each prime factor appear an even number of times. Since 2 appears 3 times, we need to add one more factor of 2. This means we want k to be 2^1 = 2.
Checking, we have:
C = 216k = 216(2) = 2^3 × 3^3 × 2 = (2^2 × 3^2)^2 = 36^2
Therefore, the least value of k that will make C a perfect square is k = 2.
To find the least value of k that will make C a perfect square, we need to find the smallest perfect square that is divisible by 216.
Step 1: Prime factorize 216.
216 = 2^3 * 3^3
Step 2: Identify the highest power of each prime factor.
Since we want to find the perfect square, the highest power of each prime factor should be even. In this case, the highest power of 2 is 2^3 and the highest power of 3 is 3^3.
Step 3: Calculate the least value of k.
To make the highest power of 2 even, we need to add one more factor of 2. Similarly, to make the highest power of 3 even, we need to add one more factor of 3.
Therefore, the least value of k is:
k = 3 + 1 = 4
So, if k = 4, C = 216k = 216 * 4 = 864, which is a perfect square since it can be expressed as 2^5 * 3^2.