Questions: Find a) the greatest common divisor (GCD) and b) the least common multiple (LCM). 63,567 , and 729 a) The GCD is

Transcript text: Find a) the greatest common divisor (GCD) and b) the least common multiple (LCM). 63,567 , and 729 a) The GCD is $\square$

Solution

Solution Steps

To find the greatest common divisor (GCD) and the least common multiple (LCM) of the numbers 63, 567, and 729, we can use Python's built-in functions from the math module. The GCD can be found using the gcd function, and the LCM can be calculated using the relationship between GCD and LCM.

Solution Approach

Use the gcd function from the math module to find the GCD of the first two numbers, then use the result to find the GCD with the third number.
Calculate the LCM using the formula: LCM(a, b) = abs(a*b) // GCD(a, b) and extend it to three numbers.

Step 1: Calculate the GCD

To find the greatest common divisor (GCD) of the numbers $ 63 $, $ 567 $, and $ 729 $, we can use the following calculations:

\[ \text{GCD}(63, 567) = 9 \] \[ \text{GCD}(9, 729) = 9 \]

Thus, the GCD of $ 63 $, $ 567 $, and $ 729 $ is $ 9 $.

Step 2: Calculate the LCM

To find the least common multiple (LCM) of the numbers $ 63 $, $ 567 $, and $ 729 $, we can use the relationship between GCD and LCM:

\[ \text{LCM}(a, b) = \frac{|a \cdot b|}{\text{GCD}(a, b)} \]

Calculating step-by-step:

Calculate $ \text{LCM}(63, 567) $: \[ \text{LCM}(63, 567) = \frac{|63 \cdot 567|}{\text{GCD}(63, 567)} = \frac{35721}{9} = 3969 \]
Now calculate $ \text{LCM}(3969, 729) $: \[ \text{LCM}(3969, 729) = \frac{|3969 \cdot 729|}{\text{GCD}(3969, 729)} = \frac{2897921}{729} = 5103 \]

Thus, the LCM of $ 63 $, $ 567 $, and $ 729 $ is $ 5103 $.