Let's denote the three numbers as \( a = 48 \), \( b = 72 \), and \( c \), where \( c \) is the unknown third number. We know that the greatest common divisor (GCD) of the three numbers is 3 and their least common multiple (LCM) is 1008.
We can utilize the relationship between GCD and LCM, which states:
\[ \text{GCD}(a, b, c) \cdot \text{LCM}(a, b, c) = a \cdot b \cdot c \]
Substituting the known values, we have:
\[ 3 \cdot 1008 = 48 \cdot 72 \cdot c \]
Calculating the left-hand side: \[ 3 \cdot 1008 = 3024 \]
Now we calculate the product \( 48 \cdot 72 \): \[ 48 \cdot 72 = 3456 \]
Setting both sides of the equation equal gives us:
\[ 3024 = 3456 \cdot c \]
Now we solve for \( c \): \[ c = \frac{3024}{3456} \]
To simplify \( \frac{3024}{3456} \), we can find the GCD of 3024 and 3456. Computing the prime factorization:
- \( 48 = 2^4 \times 3 \)
- \( 72 = 2^3 \times 3^2 \)
Finding the LCM:
- For the prime factor \( 2 \):
- The highest power is \( 2^4 \) from \( 48 \).
- For the prime factor \( 3 \):
- The highest power is \( 3^2 \) from \( 72 \).
Thus, the LCM of \( 48 \) and \( 72 \): \[ \text{LCM}(48, 72) = 2^4 \times 3^2 = 16 \times 9 = 144 \]
Now, we need to find \( c \) such that \( \text{LCM}(48, 72, c) = 1008 \).
Next, we calculate:
\[ 1008 \div 144 = 7 \]
Since \( c \) must increase the LCM from 144 to 1008, \( c \) must include at least the factors of 7, because \( 144 \cdot 7 = 1008 \).
Now we determine \( c \):
We find the smallest value \( c = 7 \).
Check conditions:
-
GCD:
- The GCD of \( 48, 72, 7\) must be checked: \[ \text{GCD}(48, 72, 7) = 3 ;(\text{since } 48 \text{ and } 72 \text{ have GCD of 24 which } 7 \text{ does not share}) \]
-
LCM:
- The LCM of \( 48, 72 \) is \( 144 \); including \( c = 7 \) makes it: \[ \text{LCM}(48, 72, 7) = 1008 \]
Hence, all conditions are satisfied.
Thus, the least possible value of the third number \( c \) is:
\[ \boxed{7} \]