To find the product of the L.C.M (Least Common Multiple) and G.C.D (Greatest Common Divisor) of the numbers 16, 24, and 30, follow these steps:
1. Find the G.C.D of the three numbers:
- The G.C.D of two numbers is the largest number that divides both of them without leaving a remainder. For three numbers, it is the largest number that divides all three numbers without a remainder.
- First, find the G.C.D of any two numbers, say 16 and 24, which is 8.
- Then, find the G.C.D of this result with the third number, which is 30. The G.C.D of 8 and 30 is 2.
- Therefore, the G.C.D of 16, 24, and 30 is 2.
2. Find the L.C.M of the three numbers:
- The L.C.M of two numbers is the smallest number that is a multiple of both of them. For three numbers, it is the smallest number that is a multiple of all three numbers.
- To find the L.C.M, we generally use the formula:
[tex]\[
\text{LCM}(a, b, c) = \text{LCM}(\text{LCM}(a, b), c)
\][/tex]
- First, find the L.C.M of any two numbers, say 16 and 24. The L.C.M of 16 and 24 is 48.
- Then, find the L.C.M of this result with the third number, which is 30. The L.C.M of 48 and 30 is 240.
- Therefore, the L.C.M of 16, 24, and 30 is 240.
3. Calculate the product of the G.C.D and L.C.M:
- The product of G.C.D and L.C.M of the numbers 16, 24, and 30 can be calculated as:
[tex]\[
\text{Product} = \text{G.C.D} \times \text{L.C.M} = 2 \times 240 = 480
\][/tex]
Therefore, the product of the L.C.M and G.C.D of 16, 24, and 30 is 480.
