GCD Euclidean Algorithm Quiz: Find the GCD Easily

Using the Euclidean Algorithm: 98 divided by 56 gives 1 remainder 42. Then 56 divided by 42 gives 1 remainder 14. Then 42 divided by 14 gives 3 remainder 0. The last nonzero remainder is 14, so GCD(56, 98) = 14. The algorithm always starts with the larger number divided by the smaller, and stops when the remainder reaches 0.

Using the Euclidean Algorithm: 180 divided by 48 gives 3 remainder 36. Then 48 divided by 36 gives 1 remainder 12. Then 36 divided by 12 gives 3 remainder 0. The last nonzero remainder is 12, so GCD(180, 48) = 12. Each step uses the previous remainder as the new divisor until the remainder reaches 0.

The answer is True. When one of the two numbers is 0, the Euclidean Algorithm gives the result immediately. Dividing a by 0 is not performed — instead, the rule states that GCD(a, 0) = a because a is the largest number that divides a, and every integer divides 0. This is a foundational property of the GCD used as the base case in the algorithm.

Using the Euclidean Algorithm: 315 divided by 135 gives 2 remainder 45. Then 135 divided by 45 gives 3 remainder 0. The last nonzero remainder is 45, so GCD(315, 135) = 45. Because the remainder reached 0 in just two steps, this is a short computation. The result of 45 means 45 is the largest number that divides both 315 and 135 exactly.

Using the Euclidean Algorithm: 91 divided by 35 gives 2 remainder 21. Then 35 divided by 21 gives 1 remainder 14. Then 21 divided by 14 gives 1 remainder 7. Then 14 divided by 7 gives 2 remainder 0. The last nonzero remainder is 7, so GCD(91, 35) = 7. This means 7 is the largest number that divides both 91 and 35 without leaving a remainder.

The GCD divides both original numbers exactly — this is the definition of a common divisor, and the algorithm finds the greatest one. Option C is also correct: the last nonzero remainder in the Euclidean Algorithm is always the GCD. Option B is wrong because the GCD equals the smaller number only when the smaller number divides the larger exactly. Option D is wrong because the GCD can be any positive integer, odd or even.

The answer is False. The Euclidean Algorithm works for any two positive integers, whether they are even, odd, or one of each. For example, GCD(91, 35) uses two odd numbers, and GCD(56, 99) uses one even and one odd number. The algorithm places no restriction on whether the inputs are even or odd. It only requires both numbers to be positive integers.

Using the Euclidean Algorithm: 169 divided by 65 gives 2 remainder 39. Then 65 divided by 39 gives 1 remainder 26. Then 39 divided by 26 gives 1 remainder 13. Then 26 divided by 13 gives 2 remainder 0. The last nonzero remainder is 13, so GCD(169, 65) = 13. Note that 169 equals 13 multiplied by 13, and 65 equals 13 multiplied by 5, confirming that 13 divides both.

Using the Euclidean Algorithm: 200 divided by 75 gives 2 remainder 50. Then 75 divided by 50 gives 1 remainder 25. Then 50 divided by 25 gives 2 remainder 0. The last nonzero remainder is 25, so GCD(200, 75) = 25. The result means 25 is the largest number that divides both 200 and 75 exactly: 200 divided by 25 equals 8, and 75 divided by 25 equals 3.

Using the Euclidean Algorithm: 78 divided by 30 gives 2 remainder 18. Then 30 divided by 18 gives 1 remainder 12. Then 18 divided by 12 gives 1 remainder 6. Then 12 divided by 6 gives 2 remainder 0. The last nonzero remainder is 6, so GCD(78, 30) = 6. The result means 6 is the largest number that divides both 78 and 30 exactly without leaving a remainder.

Using the Euclidean Algorithm: 350 divided by 140 gives 2 remainder 70. Then 140 divided by 70 gives 2 remainder 0. The last nonzero remainder is 70, so GCD(350, 140) = 70. The algorithm reached 0 in just two steps because 70 divides 140 exactly. Confirming: 350 divided by 70 equals 5, and 140 divided by 70 equals 2, so 70 divides both original numbers.

The answer is True. By definition, a common divisor of two numbers a and b is any number that divides both a and b exactly. The greatest common divisor is the largest such number. Because it is a common divisor, it must divide both a and b without leaving a remainder. This is what the Euclidean Algorithm finds, and it can always be verified by dividing both original numbers by the GCD.

Using the Euclidean Algorithm: 204 divided by 84 gives 2 remainder 36. Then 84 divided by 36 gives 2 remainder 12. Then 36 divided by 12 gives 3 remainder 0. The last nonzero remainder is 12, so GCD(204, 84) = 12. Confirming: 204 divided by 12 equals 17, and 84 divided by 12 equals 7, so 12 divides both original numbers exactly.

Step 1 is correct: 84 multiplied by 1 equals 84, and 84 plus 36 equals 120. Step 2 contains an error: 36 multiplied by 2 equals 72, and 72 plus 10 equals 82, not 84. The correct step is 84 = 36 x 2 + 12. Step 3 contains an error because it uses 10 from the wrong remainder in Step 2. Using the correct remainder of 12: 36 = 12 x 3 + 0, so the GCD is 12. Step 4 is based on the same incorrect remainder and is therefore also invalid.

Using the Euclidean Algorithm: 189 divided by 81 gives 2 remainder 27. Then 81 divided by 27 gives 3 remainder 0. The last nonzero remainder is 27, so GCD(189, 81) = 27. The algorithm completed in just two steps. Confirming: 189 divided by 27 equals 7, and 81 divided by 27 equals 3, so 27 divides both original numbers exactly.

The answer is False. When GCD(a, b) = 1, the two numbers are called coprime, meaning they share no common factor other than 1. However, neither number needs to be prime. For example, GCD(8, 9) = 1 because 8 equals 2 multiplied by 2 multiplied by 2 and 9 equals 3 multiplied by 3, and they share no common factor. Both 8 and 9 are composite, yet they are coprime.

Using the Euclidean Algorithm: 175 divided by 125 gives 1 remainder 50. Then 125 divided by 50 gives 2 remainder 25. Then 50 divided by 25 gives 2 remainder 0. The last nonzero remainder is 25, so GCD(175, 125) = 25. Confirming: 175 divided by 25 equals 7, and 125 divided by 25 equals 5, so 25 divides both original numbers exactly.

This is a GCD application. The largest square tile that fits exactly must have a side length that divides both 48 and 36. Using the Euclidean Algorithm to find GCD(48, 36): 48 divided by 36 gives 1 remainder 12; 36 divided by 12 gives 3 remainder 0. GCD = 12. A tile of 12 cm fits exactly: 48 divided by 12 equals 4 tiles across, and 36 divided by 12 equals 3 tiles down, covering the floor with 12 tiles total and no cutting needed.

The answer is False. The correct relationship is that GCD(a, b) multiplied by LCM(a, b) equals a multiplied by b, not a plus b. For example, GCD(12, 8) = 4 and LCM(12, 8) = 24. Multiplying 4 by 24 gives 96, which equals 12 multiplied by 8. Adding 12 and 8 gives only 20, which is not equal to 96. The product formula, not the sum, is the correct relationship.

To find the GCD of three numbers, first find GCD(24, 60), then apply the algorithm to that result and 84. GCD(24, 60): 60 divided by 24 gives 2 remainder 12; 24 divided by 12 gives 2 remainder 0. GCD = 12. Then GCD(12, 84): 84 divided by 12 gives 7 remainder 0. GCD = 12. So GCD(24, 60, 84) = 12. Confirming: 12 divides 24, 60, and 84 exactly, and no number larger than 12 divides all three.