Questions: lcm(51,68) and gcd(51,68)

Transcript text: $\operatorname{lcm}(51,68)$ and $\operatorname{gcd}(51,68)$

Solution

Solution Steps

To find the least common multiple (LCM) and greatest common divisor (GCD) of two numbers, we can use the relationship between them. The product of the LCM and GCD of two numbers is equal to the product of the numbers themselves. We can use Python's built-in functions to compute these values directly.

Step 1: Calculate GCD

To find the greatest common divisor (GCD) of $ 51 $ and $ 68 $, we can use the Euclidean algorithm. The GCD is calculated as follows: \[ \gcd(51, 68) = 17 \]

Step 2: Calculate LCM

The least common multiple (LCM) can be calculated using the relationship: \[ \operatorname{lcm}(a, b) \cdot \operatorname{gcd}(a, b) = a \cdot b \] Substituting the known values: \[ \operatorname{lcm}(51, 68) = \frac{51 \cdot 68}{\gcd(51, 68)} = \frac{51 \cdot 68}{17} = 204 \]

Final Answer

The results are: \[ \gcd(51, 68) = 17 \] \[ \operatorname{lcm}(51, 68) = 204 \] Thus, the final answers are: \[ \boxed{\gcd = 17} \] \[ \boxed{\operatorname{lcm} = 204} \]

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