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 $\text{GCD} \times \text{LCM} = \text{Product of the numbers}$.