a)

To determine the greatest common divisor (gcd) of 72345 and 43215 by using the Euclidean algorithm:

Step 1: Divide 72345 by 43215, and obtain the remainder.
        72345 ÷ 43215 = 1 remainder 29130

Step 2: Divide 43215 by 29130, and obtain the remainder.
        43215 ÷ 29130 = 1 remainder 14085

Step 3: Divide 29130 by 14085, and obtain the remainder.
        29130 ÷ 14085 = 2 remainder 945

Step 4: Divide 14085 by 945, and obtain the remainder.
        14085 ÷ 945 = 14 remainder 615

Step 5: Divide 945 by 615, and obtain the remainder.
        945 ÷ 615 = 1 remainder 330

Step 6: Divide 615 by 330, and obtain the remainder.
        615 ÷ 330 = 1 remainder 285

Step 7: Divide 330 by 285, and obtain the remainder.
        330 ÷ 285 = 1 remainder 45

Step 8: Divide 285 by 45, and obtain the remainder.
        285 ÷ 45 = 6 remainder 15

Step 9: Divide 45 by 15, and obtain the remainder.
        45 ÷ 15 = 3 remainder 0

Step 10: Since the remainder is now 0, we stop.

The gcd(72345, 43215) is the last non-zero remainder, which is 15.

Therefore, gcd(72345, 43215) = 15.

b)

 To determine the gcd of 3486 and 10292 by using  the Euclidean algorithm:

Step 1: Divide 10292 by 3486, and obtain the remainder.
        10292 ÷ 3486 = 2 remainder 3320

Step 2: Divide 3486 by 3320, and obtain the remainder.
        3486 ÷ 3320 = 1 remainder 166

Step 3: Divide 3320 by 166, and obtain the remainder.
        3320 ÷ 166 = 20 remainder 120

Step 4: Divide 166 by 120, and obtain the remainder.
        166 ÷ 120 = 1 remainder 46

Step 5: Divide 120 by 46, and obtain the remainder.
        120 ÷ 46 = 2 remainder 28

Step 6: Divide 46 by 28, and obtain the remainder.
        46 ÷ 28 = 1 remainder 18

Step 7: Divide 28 by 18, and obtain the remainder.
        28 ÷ 18 = 1 remainder 10

Step 8: Divide 18 by 10, and obtain the remainder.
        18 ÷ 10 = 1 remainder 8

Step 9: Divide 10 by 8, and obtain the remainder.
        10 ÷ 8 = 1 remainder 2

Step 10: Divide 8 by 2, and obtain the remainder.
        8 ÷ 2 = 4 remainder 0

Since the remainder is now 0, we stop.

The gcd(3486, 10292) is the last non-zero remainder, which is 2.

Therefore, gcd(3486, 10292) = 2.