Pipeline Alpha alone can complete a data migration task in 10 hours, Pipeline Beta alone can complete it in 15 hours, and Pipeline Gamma alone can complete it in N hours. All three pipelines worked together on the task for 3 hours, after which Pipeline Alpha was disconnected. The remaining work was completed by Pipeline Beta and Pipeline Gamma together in 4 hours. Find the value of N.

The correct answer is 30.

Step 1: Determine the work rates of each pipeline.

Let the total data migration task be represented as 1 complete unit of work.

Work rate of Pipeline Alpha = $\frac{1}{10}$ task per hour.

Work rate of Pipeline Beta = $\frac{1}{15}$ task per hour.

Work rate of Pipeline Gamma = $\frac{1}{N}$ task per hour.

Step 2: Calculate the work done in the first 3 hours by all three pipelines together.

Combined hourly rate of Alpha, Beta, and Gamma = $\frac{1}{10}+\frac{1}{15}+\frac{1}{N}$

First, sum the rates of Alpha and Beta:

$\frac{1}{10}+\frac{1}{15}=\frac{3+2}{30}=\frac{5}{30}=\frac{1}{6}$

Therefore, the total work completed in 3 hours is:

$\text{Work done in 3 hours}=3\times \left(\frac{1}{6}+\frac{1}{N}\right)=\frac{1}{2}+\frac{3}{N}$

Step 3: Calculate the remaining work to be completed.

$\text{Remaining work}=1-\left(\frac{1}{2}+\frac{3}{N}\right)=\frac{1}{2}-\frac{3}{N}$

Step 4: Calculate the work done by Pipeline Beta and Pipeline Gamma in 4 hours.

After Pipeline Alpha is disconnected, Pipeline Beta and Pipeline Gamma complete the remaining work together in 4 hours.

Combined hourly rate of Beta and Gamma = $\frac{1}{15}+\frac{1}{N}$

Work done in 4 hours = $4\times \left(\frac{1}{15}+\frac{1}{N}\right)=\frac{4}{15}+\frac{4}{N}$

Step 5: Solve for N.

Equate the remaining work to the work done by Beta and Gamma in 4 hours:

$\frac{1}{2}-\frac{3}{N}=\frac{4}{15}+\frac{4}{N}$

Rearrange the equation to isolate the terms containing $N$:

$\frac{1}{2}-\frac{4}{15}=\frac{4}{N}+\frac{3}{N}$

Find a common denominator for the left side (30):

$\frac{15}{30}-\frac{8}{30}=\frac{7}{N}$

$\frac{7}{30}=\frac{7}{N}$

Equating the denominators yields:

$N=30$

Hence, the value of N is 30.