The least common multiple of two numbers is 3780, and the greatest common divisor is 18. Given that one of the numbers is 180, what is the other number?
We use the identity $\gcd(a,b) \cdot \mathop{\text{lcm}}[a,b] = ab$ for all positive integers $a$ and $b$.  We are told that $\gcd(a,b) = 18$, and $\mathop{\text{lcm}}[a,b] = 3780$, so $ab = 18 \cdot 3780$.  If one number is 180, then the other number is $18 \cdot 3780/180 = \boxed{378}$.
378
Level 4
Number Theory
Problem: The least common multiple of two numbers is 3780, and the greatest common divisor is 18. Given that one of the numbers is 180, what is the other number?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the other number, we can use the relationship between the least common multiple (LCM) and the greatest common divisor (GCD) of two numbers. The formula connecting these values is:

LCM(a, b) * GCD(a, b) = a * b

Here, let a be 180 (the known number), and let b be the unknown number we want to find. We know:

LCM(180, b) = 3780
GCD(180, b) = 18

Substituting the known values into the formula gives:

3780 * 18 = 180 * b

Calculating the left side:

3780 * 18 = 68040

Now we can set up the equation:

68040 = 180 * b

To isolate b, we divide both sides by 180:

b = 68040 / 180

Calculating the division:

68040 / 180 = 378

Thus, the other number is 378.<|END_OF_LATENT|> The final answer is $378$.