Calculating L.C.M. of Time Durations

To find the Least Common Multiple (L.C.M.) of the given time durations, we first convert them into a single unit, minutes.

Step 1: Convert Time Durations to Minutes

• Duration 1: $15$ hours $36$ mins.
  • Calculation: $(15 \times 60) + 36 = 900 + 36 = 936$ minutes.
• Duration 2: $6$ hours $16$ mins.
  • Calculation: $(6 \times 60) + 16 = 360 + 16 = 376$ minutes.

Step 2: Calculate the L.C.M. in Minutes

Now, we find the L.C.M. of $936$ and $376$ minutes.

Prime factorization:

• $936 = 2^3 \times 3^2 \times 13$
• $376 = 2^3 \times 47$

The L.C.M. is the product of the highest powers of all prime factors involved:

L.C.M. $= 2^3 \times 3^2 \times 13 \times 47 = 8 \times 9 \times 13 \times 47 = 43992$ minutes.

Step 3: Convert L.C.M. back to Hours and Minutes

To convert $43992$ minutes back into hours and minutes, we divide by $60$:

$\frac{43992}{60} = 733.2$ hours.

This means $733$ whole hours and $0.2$ of an hour.

Convert the fractional part back to minutes:

$0.2 \times 60 = 12$ minutes.

Therefore, the L.C.M. is $733$ hours and $12$ minutes.