Find each of the following.

(a) GCD(12, 26, 65)

(b) LCM(12, 26, 65)

the GCD for 12,26,65 is 1

that is, there is no divisor or common factor other than 1 which divides into all 3 numbers

for LCM
12 = 2x6
26 = 2x13
65 = 5x13

so we need 2x6x13x5 or 780

To find the greatest common divisor (GCD) of three or more numbers, you can use the method of finding the GCD of two numbers and then iteratively finding the GCD with the next number. Similarly, to find the least common multiple (LCM), you can find the LCM of two numbers and then iteratively find the LCM with the next number.

(a) GCD(12, 26, 65):

Step 1: Find the GCD of the first two numbers, 12 and 26.
To find the GCD of 12 and 26, you can use the Euclidean algorithm. Divide the larger number (26) by the smaller number (12), and take the remainder.
26 ÷ 12 = 2 remainder 2

Step 2: Repeat the process with the remainder and the next number.
Now, find the GCD of 12 and 2.
12 ÷ 2 = 6 remainder 0

Since the remainder is 0, the GCD of 12 and 26 is 2.

Step 3: Find the GCD of the obtained result (2) and the next number, 65.
To find the GCD of 2 and 65, apply the Euclidean algorithm.
65 ÷ 2 = 32 remainder 1
2 ÷ 1 = 2 remainder 0

The remainder is 0, so the GCD of 2 and 65 is 1.

The GCD of 12, 26, and 65 is 1.

(b) LCM(12, 26, 65):

Step 1: Find the LCM of the first two numbers, 12 and 26.
To find the LCM, divide the product of the two numbers by their GCD.
LCM(12, 26) = (12 × 26) / GCD(12, 26)

We already found that GCD(12, 26) = 2.
So, LCM(12, 26) = (12 × 26) / 2 = 312 / 2 = 156.

Step 2: Repeat the process with the LCM obtained above and the next number, 65.
LCM(156, 65) = (156 × 65) / GCD(156, 65)

To find the GCD of 156 and 65, apply the Euclidean algorithm.
156 ÷ 65 = 2 remainder 26
65 ÷ 26 = 2 remainder 13
26 ÷ 13 = 2 remainder 0

The GCD of 156 and 65 is 13.

So, LCM(156, 65) = (156 × 65) / 13 = 10140 / 13 = 780.

Therefore, the LCM of 12, 26, and 65 is 780.