https://r2.erweima.ai/mathbot/image/1725778114650bgj82td7.jpg

The problem in the image is written in Chinese. Here's a translation and breakdown:

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### Problem Description:
Given a positive integer \( a \), please find three positive integers \( b, c, d \) such that:
\[
a + b + c + d = \gcd(a, b) + \text{lcm}(c, d),
\]
where \( \gcd \) is the greatest common divisor and \( \text{lcm} \) is the least common multiple.

Additionally, the condition is that the combined value of \( b, c, d \) must satisfy:
\[
b + c + d \leq 1,634,826,193.
\]

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This problem is asking for integers \( b, c, d \) that fulfill the given equation involving both the GCD and LCM, under the sum constraint.

Would you like to discuss strategies to solve this problem, or focus on a specific part of it?

### Five Related Questions:
1. What properties of \( \gcd \) and \( \text{lcm} \) can we use to solve such equations?
2. How can we approach finding \( b, c, d \) given an arbitrary \( a \)?
3. Are there specific algorithms to compute \( \gcd \) and \( \text{lcm} \) efficiently for large numbers?
4. What role does the condition \( b + c + d \leq 1,634,826,193 \) play in this problem?
5. How does the relationship between \( a \), \( b \), and \( d \) influence possible values of \( c \)?

### Tip:
Always remember the identity: \( \gcd(x, y) \times \text{lcm}(x, y) = x \times y \). It often helps simplify such problems.

Given a positive integer a, find three positive integers b, c, d such that a + b + c + d = gcd(a, b) + lcm(c, d) and b + c + d ≤ 1,634,826,193.

Explore how to find three integers b, c, and d such that they satisfy the equation a + b + c + d = gcd(a, b) + lcm(c, d), where b + c + d is limited to 1,634,826,193. Learn about key concepts in number theory, including GCD and LCM properties.

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