The *GCD*(*a*, *b*) = 9, the *LCM*(*a*, *b*) = 378. Find the least possible value of *a*+*b*.

We are given:

- GCD(a, b) = 9
- LCM(a, b) = 378

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

### Key Formula
The relationship between GCD, LCM, and the product of two numbers is given by the formula:

\[
\text{GCD}(a, b) \times \text{LCM}(a, b) = a \times b
\]

Substituting the given values:

\[
9 \times 378 = a \times b
\]

This simplifies to:

\[
a \times b = 3402
\]

### Step 1: Factorizing 3402
We need to factorize 3402 to find two numbers \( a \) and \( b \) whose GCD is 9. First, let's find the prime factorization of 3402:

\[
3402 \div 2 = 1701
\]
\[
1701 \div 3 = 567
\]
\[
567 \div 3 = 189
\]
\[
189 \div 3 = 63
\]
\[
63 \div 3 = 21
\]
\[
21 \div 3 = 7
\]
\[
7 \div 7 = 1
\]

Thus, the prime factorization of 3402 is:

\[
3402 = 2 \times 3^5 \times 7
\]

### Step 2: Finding \( a \) and \( b \) with GCD of 9
We want \( a \) and \( b \) such that their GCD is 9. Since GCD(a, b) = 9, both \( a \) and \( b \) must be divisible by 9. Therefore, we can write \( a = 9m \) and \( b = 9n \), where \( \text{GCD}(m, n) = 1 \).

Substituting into the product equation \( a \times b = 3402 \):

\[
(9m) \times (9n) = 3402
\]
\[
81mn = 3402
\]
\[
mn = \frac{3402}{81} = 42
\]

So, we need to find pairs \( (m, n) \) such that \( mn = 42 \) and \( \text{GCD}(m, n) = 1 \).

### Step 3: Finding pairs \( (m, n) \)
The factor pairs of 42 are:

\[
(1, 42), (2, 21), (3, 14), (6, 7)
\]

We now check the GCD of each pair:
- \( \text{GCD}(1, 42) = 1 \)
- \( \text{GCD}(2, 21) = 1 \)
- \( \text{GCD}(3, 14) = 1 \)
- \( \text{GCD}(6, 7) = 1 \)

All pairs have GCD 1, so any pair works.

### Step 4: Minimizing \( a + b \)
Now we calculate \( a \) and \( b \) for each pair, using \( a = 9m \) and \( b = 9n \):
- For \( (m, n) = (1, 42) \), \( a = 9 \times 1 = 9 \), \( b = 9 \times 42 = 378 \), so \( a + b = 9 + 378 = 387 \).
- For \( (m, n) = (2, 21) \), \( a = 9 \times 2 = 18 \), \( b = 9 \times 21 = 189 \), so \( a + b = 18 + 189 = 207 \).
- For \( (m, n) = (3, 14) \), \( a = 9 \times 3 = 27 \), \( b = 9 \times 14 = 126 \), so \( a + b = 27 + 126 = 153 \).
- For \( (m, n) = (6, 7) \), \( a = 9 \times 6 = 54 \), \( b = 9 \times 7 = 63 \), so \( a + b = 54 + 63 = 117 \).

### Step 5: Conclusion
The least possible value of \( a + b \) is **117**.

Would you like any further details or have any questions?

Here are five related questions for you to explore:
1. How can you use prime factorizations to find the GCD and LCM of two numbers?
2. What is the significance of the relationship \( \text{GCD}(a, b) \times \text{LCM}(a, b) = a \times b \)?
3. Can you find two numbers whose GCD is 6 and LCM is 120? What is \( a + b \)?
4. How does changing the GCD affect the product of two numbers?
5. How can the Euclidean algorithm help in finding the GCD of two numbers?

**Tip:** When solving GCD and LCM problems, always remember the formula \( \text{GCD} \times \text{LCM} = \text{Product of the numbers} \).

The GCD(a, b) = 9, the LCM(a, b) = 378. Find the least possible value of a + b.

Explore how to find the least possible value of a + b when given GCD(a, b) = 9 and LCM(a, b) = 378. This step-by-step solution uses number theory concepts like prime factorization and the GCD-LCM product theorem. Suitable for students in Grades 8-10.

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