Question

The GCD and LCM of three numbers is 3 and 1008. If two of the numbers are 48 and 72 respectively, find the least possible value of the third number

Answer 1

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:
1. For the prime factor \( 2 \):
- The highest power is \( 2^4 \) from \( 48 \).
2. 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:**
1. **GCD**:
- The GCD of \( 48, 72, 7\) must be checked:
\[
\text{GCD}(48, 72, 7) = 3 \;(\text{since } 48 \text{ and } 72 \text{ have GCD of 24 which } 7 \text{ does not share})
\]

2. **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:

\[
\boxed{7}
\]