The LCM of three numbers is 7920 and their GCD is 12. Two of the numbers are 48 and 264. Using factor method determine the third least possible number
We start by factoring the given numbers:
- 48 factors into $2^4 \cdot 3$
- 264 factors into $2^3 \cdot 3 \cdot 11$
We know that the GCD of these two numbers is 12, which means that their common factors must include $2^2$ and $3$. To find the third possible number, we need to look for a factorization that includes these common factors and also has a least common multiple of 7920.
We begin by writing 7920 as a product of its prime factors:
$$7920 = 2^4 \cdot 3^2 \cdot 5 \cdot 11$$
To get a factorization that includes $2^2$ and $3$, we can take out one $2$ and one $3$. We are left with:
$$7920 = 2^3 \cdot 3 \cdot 5 \cdot 11 \cdot (2 \cdot 3)$$
The factors $2^3 \cdot 3$ and $2 \cdot 3$ represent the given numbers 48 and 264, respectively. To find the third possible number, we need to divide out the factors that we have used so far. That is, we divide out one factor of $2$, one factor of $3$, and one factor of $11$:
$$\frac{7920}{2 \cdot 3 \cdot 11} = 120$$
So the third possible number that satisfies the given conditions is 120. Note that there may be other numbers that also work, but 120 is the least possible one.