When the least common multiple of two positive integers is divided by their greatest common divisor, the result is 33. If one integer is 45, what is the smallest possible value of the other integer?
Let $n$ be the other integer, so \[\frac{\mathop{\text{lcm}}[45,n]}{\gcd(45,n)} = 33.\]We know that $\gcd(m,n) \cdot \mathop{\text{lcm}}[m,n] = mn$ for all positive integers $m$ and $n$, so \[\gcd(45,n) \cdot \mathop{\text{lcm}}[45,n] = 45n.\]Dividing this equation by the previous equation, we get \[[\gcd(45,n)]^2 = \frac{45n}{33} = \frac{15n}{11},\]so $11 [\gcd(45,n)]^2 = 15n$.

Since 11 divides the left-hand side, 11 also divides the right-hand side, which means $n$ is divisible by 11.  Also, 15 divides the right-hand side, so 15 divides the left-hand side, which means $\gcd(45,n)$ is divisible by 15.  Since $45 = 3 \cdot 15$, $n$ is divisible by 15.  Hence, $n$ must be divisible by $11 \cdot 15 = 165$.

Note that $\gcd(45,165) = 15$ and $\mathop{\text{lcm}}[45,165] = 495$, and $495/15 = 33$, so $n=165$ is achievable and the smallest possible value of $n$ is $\boxed{165}$.
165
Level 5
Number Theory
Problem: When the least common multiple of two positive integers is divided by their greatest common divisor, the result is 33. If one integer is 45, what is the smallest possible value of the other integer?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To solve the problem, we start by recalling the relationship between the least common multiple (LCM) and the greatest common divisor (GCD) of two integers. The formula is given by:

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

Let the two integers be a and b, where a = 45 and b is the unknown integer. We denote GCD(45, b) as g and LCM(45, b) as l. According to the problem, we have:

l / g = 33

From the relationship, we can express LCM in terms of GCD:

l = (a * b) / g

Substituting this into the equation gives:

(a * b) / g^2 = 33

Now substituting a = 45 into the equation:

(45 * b) / g^2 = 33

Rearranging this leads to:

45 * b = 33 * g^2

Thus, we can express b as:

b = (33 * g^2) / 45

To find the smallest possible value of b, we need to determine the possible values of g, which is the GCD of 45 and b. The divisors of 45 are 1, 3, 5, 9, 15, and 45. We will evaluate each divisor to find the corresponding value of b.

1. If g = 1:
   b = (33 * 1^2) / 45 = 33 / 45 = 0.733 (not an integer)

2. If g = 3:
   b = (33 * 3^2) / 45 = (33 * 9) / 45 = 297 / 45 = 6.6 (not an integer)

3. If g = 5:
   b = (33 * 5^2) / 45 = (33 * 25) / 45 = 825 / 45 = 18.33 (not an integer)

4. If g = 9:
   b = (33 * 9^2) / 45 = (33 * 81) / 45 = 2673 / 45 = 59.4 (not an integer)

5. If g = 15:
   b = (33 * 15^2) / 45 = (33 * 225) / 45 = 7425 / 45 = 165 (an integer)

6. If g = 45:
   b = (33 * 45^2) / 45 = 33 * 45 = 1485 (an integer, but larger than 165)

The smallest integer value of b occurs when g = 15, yielding b = 165. Thus, the smallest possible value of the other integer is 165.<|END_OF_LATENT|> The final answer is $165$.