Suppose LCM(a, b) = 20160, a= 60. Find the least possible positive value of b.

We are given that:

\[
\text{LCM}(a, b) = 20160
\]
and
\[
a = 60.
\]

We need to find the least possible positive value of \( b \).

### Step 1: Prime Factorization of 60

We start by finding the prime factorization of \( a = 60 \).

\[
60 = 2^2 \times 3 \times 5
\]

### Step 2: Prime Factorization of 20160

Next, we find the prime factorization of 20160. Performing prime factorization gives:

\[
20160 = 2^6 \times 3^2 \times 5 \times 7
\]

### Step 3: LCM Formula

The formula for the LCM of two numbers \( a \) and \( b \) is:

\[
\text{LCM}(a, b) = \prod \text{max}(p^i, p^j)
\]

where \( p^i \) and \( p^j \) are the powers of prime \( p \) in \( a \) and \( b \), respectively.

### Step 4: Deductions from the LCM

We already know that the LCM of \( a = 60 \) and \( b \) is 20160. Using the prime factorizations of \( 60 \) and \( 20160 \), we compare the powers of primes:

- For \( 2 \), the LCM has \( 2^6 \). Since \( a \) has \( 2^2 \), \( b \) must have at least \( 2^6 \).
- For \( 3 \), the LCM has \( 3^2 \). Since \( a \) has \( 3^1 \), \( b \) must have at least \( 3^2 \).
- For \( 5 \), the LCM has \( 5^1 \). Since \( a \) already has \( 5^1 \), \( b \) does not need any higher power of 5.
- For \( 7 \), the LCM has \( 7^1 \). Since \( a \) has no factor of \( 7 \), \( b \) must have \( 7^1 \).

### Step 5: Prime Factorization of \( b \)

From the deductions above, the smallest \( b \) must have:

\[
b = 2^6 \times 3^2 \times 7
\]

### Step 6: Calculating \( b \)

Now, let's calculate \( b \):

\[
b = 64 \times 9 \times 7 = 4032.
\]

Thus, the least possible positive value of \( b \) is \( \boxed{4032} \).

Would you like more details on any step? Here are five related questions to deepen understanding:

1. How do you perform prime factorization of large numbers efficiently?
2. What is the relationship between GCD and LCM?
3. How can you verify the LCM of two numbers using their prime factorizations?
4. Can the LCM of two numbers be smaller than either of the numbers?
5. How does changing one factor of a number affect its LCM with another number?

**Tip:** Always verify your final answer by recalculating the LCM of the two numbers to ensure correctness.

Learn how to find the least possible positive value of b given that LCM(60, b) = 20160. This problem uses prime factorization and the LCM formula to derive b = 4032. Suitable for students in Grades 8-10.

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