What is the largest value among $\operatorname{lcm}[12,2],$ $\operatorname{lcm}[12,4],$ $\operatorname{lcm}[12,6],$ $\operatorname{lcm}[12,8],$ $\operatorname{lcm}[12,10],$ and $\operatorname{lcm}[12,12]?$ Express your answer as an integer.
The least common multiple (LCM) of two numbers is the smallest positive integer that is a multiple of both numbers. To find the LCM of two numbers, I can use prime factorization and choose the highest power of each prime factor that appears in either number. For example, $\operatorname{lcm}[12,2] = 2^2 \cdot 3^1 = 12,$ since $12 = 2^2 \cdot 3^1$ and $2 = 2^1 \cdot 3^0.$ Similarly, I can find the other LCMs by factoring both numbers and choosing the highest power of each prime factor. $\operatorname{lcm}[12,4] = 2^2 \cdot 3^1 = 12,$ since $4 = 2^2 \cdot 3^0.$ $\operatorname{lcm}[12,6] = 2^2 \cdot 3^1 = 12,$ since $6 = 2^1 \cdot 3^1.$ $\operatorname{lcm}[12,8] = 2^3 \cdot 3^1 = 24,$ since $8 = 2^3 \cdot 3^0.$ $\operatorname{lcm}[12,10] = 2^2 \cdot 3^1 \cdot 5^1 = 60,$ since $10 = 2^1 \cdot 5^1.$ $\operatorname{lcm}[12,12] = 2^2 \cdot 3^1 = 12,$ since $12 = 12.$ The largest value among these LCMs is 60, which is the LCM of 12 and 10. 