Solution: To find when both projects synchronize, compute the least common multiple (LCM) of 10 and 15. Factorize: $10 = 2 \times 5$ and $15 = 3 \times 5$. The LCM is $2 \times 3 \times 5 = 30$. Thus, the projects will align after $\boxed{30}$ months.

["# Finding When Synchronized Projects Align: Using LCM to Solve the Problem", "When managing multiple projects that follow periodic cycles, one key question often arises: When will both projects synchronize again? This common challenge appears in engineering, scheduling, and teaching mathematical concepts—especially around time intervals and cycles. The solution lies in computing the Least Common Multiple (LCM), a powerful tool that reveals when synchronized events will coincide.", "## Understanding the Problem", "Imagine two projects that complete cycles every 10 months and 15 months, respectively. To determine when both will finish simultaneously, we must find the earliest time unit that is a multiple of both 10 and 15. This is precisely what the Least Common Multiple provides.", "## The Math Behind Synchronization: Factorization", "To compute the LCM, we begin by factorizing each number into its prime components:", "- $10 = 2 \ imes 5$\n- $15 = 3 \ imes 5$", "The LCM is obtained by taking the highest power of every prime factor that appears in either factorization. Since:", "- $2$ appears only in 10 as $2^1$\n- $3$ appears only in 15 as $3^1$\n- $5$ appears in both as $5^1$", "We multiply these together:", "$$\n\ ext{LCM} = 2^1 \ imes 3^1 \ imes 5^1 = 2 \ imes 3 \ imes 5 = 30\n$$", "Thus, the two projects will next synchronize their completion cycles after 30 months.", "## How This Applies to Real-World Scheduling", "This principle applies widely in manufacturing timelines, software deployment, and event planning. For example:", "- A system that runs weekly (7-day cycle) and monthly (30-day cycle) finds alignment in $LCM(7, 30) = 210$ days.\n- In agile development, sprints lasting 10 days and 15 days will align every 30 days, helping teams coordinate reviews and demos.", "### Final Answer", "The projects synchronize after $\boxed{30}$ months—the least common multiple of 10 and 15.", "Understanding and computing the LCM allows teams to plan effectively, reduce idle time, and ensure coordinated progress toward shared milestones. Whether in construction timelines or software versioning, leveraging mathematical synchronization ensures efficiency and alignment."]