What is the smallest positive integer that can be written in the form $2002m + 44444n$, where $m$ and $n$ are integers?
Notice that the question is essentially asking us for the greatest common divisor of $2002$ and $44444$: any number that can be written in the given form must be divisible by the greatest common divisor of $2002$ and $44444$. Conversely, we can find the values of $m$ and $n$ through repeated applications of the Euclidean algorithm. In particular, \begin{align*}
&\text{gcd}\,(2002, 44444) \\
&\qquad= \text{gcd}\,(2002, 44444 - 22 \cdot 2002)\\&\qquad = \text{gcd}\,(2002, 400) \\
&\qquad= \text{gcd}\,(2002 - 5 \cdot (44444 - 22 \cdot 2002), 400) \\&\qquad= \text{gcd}\,(2, 400) \\
&\qquad= \boxed{2}.
\end{align*}Notice that \begin{align*}
&2002 - 5 \cdot (44444 - 22 \cdot 2002)\\ &\qquad= 2002 - 5 \cdot 44444 + 110 \cdot 2002 \\ &\qquad= (111) \cdot 2002 + (-5) \cdot 44444 \\ &\qquad= 2,\end{align*}as desired.
2
Level 5
Number Theory
Problem: What is the smallest positive integer that can be written in the form $2002m + 44444n$, where $m$ and $n$ are integers?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the smallest positive integer expressible as $2002m + 44444n$, we need to determine the greatest common divisor (gcd) of the coefficients 2002 and 44444. The gcd will help us identify the smallest integer that can be formed by linear combinations of these two numbers.

First, we calculate the gcd using the Euclidean algorithm. We start by dividing 44444 by 2002:

1. Calculate $44444 \div 2002 \approx 22.2$, so the integer part is 22.
2. Multiply: $2002 \times 22 = 44044$.
3. Subtract: $44444 - 44044 = 400$.

Next, we apply the Euclidean algorithm again with 2002 and 400:

1. Calculate $2002 \div 400 \approx 5.005$, so the integer part is 5.
2. Multiply: $400 \times 5 = 2000$.
3. Subtract: $2002 - 2000 = 2$.

Now we apply the algorithm one more time with 400 and 2:

1. Calculate $400 \div 2 = 200$ with no remainder.

Since the remainder is now 0, we conclude that the gcd of 2002 and 44444 is 2. 

This means that any integer expressible in the form $2002m + 44444n$ must be a multiple of 2. To find the smallest positive integer, we consider the smallest positive multiple of 2, which is 2 itself.

Thus, the smallest positive integer that can be expressed in the form $2002m + 44444n$ is indeed 2.<|END_OF_LATENT|> The final answer is $2$.