GCD LCM Word Problems Quiz: Real-Life Applications

To find when both ferries next depart together, compute LCM(9, 15). Using prime factorization: 9 equals 3 squared and 15 equals 3 multiplied by 5. The LCM takes the highest power of each prime: 3 squared multiplied by 5 equals 45. Both ferries will next depart together 45 minutes after 8:00 am, at 8:45 am.

The greatest piece length that divides both 52 and 78 exactly is GCD(52, 78). Using the Euclidean Algorithm: 78 divided by 52 gives remainder 26; 52 divided by 26 gives remainder 0. GCD equals 26 metres. Confirming: 52 divided by 26 equals 2 pieces, and 78 divided by 26 equals 3 pieces, with no rope left over.

The answer is True. The LCM of a and b is the smallest positive integer divisible by both a and b. Since it must be divisible by a, it must be at least as large as a. Since it must also be divisible by b, it must be at least as large as b. Therefore LCM(a, b) is always greater than or equal to both a and b. It equals one of the numbers only when that number is itself a multiple of the other.

The greatest number of identical bouquets is GCD(75, 100). Using prime factorization: 75 equals 3 multiplied by 5 squared, and 100 equals 2 squared multiplied by 5 squared. The common prime factors are 5 squared equals 25. So GCD equals 25 bouquets. Each bouquet contains 75 divided by 25 equals 3 roses and 100 divided by 25 equals 4 tulips.

To find when both machines next align, compute LCM(8, 14). Using prime factorization: 8 equals 2 cubed and 14 equals 2 multiplied by 7. The LCM takes the highest power of each prime: 2 cubed multiplied by 7 equals 56. Both machines will next complete a cycle together after 56 minutes.

GCD(60, 84): 84 divided by 60 gives remainder 24; 60 divided by 24 gives remainder 12; 24 divided by 12 gives remainder 0. GCD equals 12. GCD(72, 84): 84 divided by 72 gives remainder 12; 72 divided by 12 gives remainder 0. GCD equals 12. GCD(48, 80): 48 equals 2 to the fourth multiplied by 3 and 80 equals 2 to the fourth multiplied by 5. GCD equals 2 to the fourth equals 16, not 12. GCD(60, 72): 72 divided by 60 gives remainder 12; 60 divided by 12 gives remainder 0. GCD equals 12. Options A, B, and D all have GCD equal to 12.

To find when both satellites next align, compute LCM(16, 20). Using prime factorization: 16 equals 2 to the fourth and 20 equals 2 squared multiplied by 5. The LCM takes the highest power of each prime: 2 to the fourth multiplied by 5 equals 80. Both satellites will next align after 80 days.

The answer is False. Two numbers whose GCD equals 1 are called relatively prime or coprime. For example, GCD(8, 15) equals 1 because 8 equals 2 cubed and 15 equals 3 multiplied by 5, and they share no common prime factor. Many pairs of numbers are coprime, so GCD equals 1 is a valid and common result. The GCD can be any positive integer from 1 up to the smaller of the two numbers.

The greatest number of identical bags is GCD(110, 154). Using the Euclidean Algorithm: 154 divided by 110 gives remainder 44; 110 divided by 44 gives remainder 22; 44 divided by 22 gives remainder 0. GCD equals 22. Each apple bag contains 110 divided by 22 equals 5 apples, and each orange bag contains 154 divided by 22 equals 7 oranges.

Compute LCM(6, 10, 15). Using prime factorization: 6 equals 2 multiplied by 3; 10 equals 2 multiplied by 5; 15 equals 3 multiplied by 5. Taking the highest power of each prime: 2 multiplied by 3 multiplied by 5 equals 30. All three runners will next complete a lap together after 30 minutes.

Compute LCM(8, 12, 18). Using prime factorization: 8 equals 2 cubed; 12 equals 2 squared multiplied by 3; 18 equals 2 multiplied by 3 squared. Taking the highest power of each prime: 2 cubed multiplied by 3 squared equals 8 multiplied by 9 equals 72. All three signals will next turn green together after 72 seconds.

The relationship between GCD, LCM, and the product of two numbers is: a multiplied by b equals GCD(a, b) multiplied by LCM(a, b). Substituting the given values: a multiplied by b equals 8 multiplied by 56 equals 448. This identity holds for any two positive integers and is a direct consequence of how prime factorizations combine in GCD and LCM calculations.

The answer is True. The identity a multiplied by b equals GCD(a, b) multiplied by LCM(a, b) always holds. If LCM(a, b) equals a multiplied by b, then substituting gives a multiplied by b equals GCD(a, b) multiplied by a multiplied by b. Dividing both sides by a multiplied by b gives GCD(a, b) equals 1. This means the two numbers share no common factor other than 1 and are therefore relatively prime.

The largest square tile must have a side length equal to GCD(144, 96). Using the Euclidean Algorithm: 144 divided by 96 gives remainder 48; 96 divided by 48 gives remainder 0. GCD equals 48 cm. Confirming: 144 divided by 48 equals 3 tiles across, and 96 divided by 48 equals 2 tiles down, covering the floor with 6 tiles total and no cutting needed.

GCD(40, 56): 56 divided by 40 gives remainder 16; 40 divided by 16 gives remainder 8; 16 divided by 8 gives remainder 0. GCD equals 8. GCD(24, 40): 40 divided by 24 gives remainder 16; 24 divided by 16 gives remainder 8; 16 divided by 8 gives remainder 0. GCD equals 8. GCD(32, 52): 32 equals 2 to the fifth and 52 equals 2 squared multiplied by 13. GCD equals 2 squared equals 4, not 8. GCD(56, 72): 72 divided by 56 gives remainder 16; 56 divided by 16 gives remainder 8; 16 divided by 8 gives remainder 0. GCD equals 8. Options A, B, and D all have GCD equal to 8.

The greatest row size that divides both 252 and 180 exactly is GCD(252, 180). Using the Euclidean Algorithm: 252 divided by 180 gives remainder 72; 180 divided by 72 gives remainder 36; 72 divided by 36 gives remainder 0. GCD equals 36. Confirming: 252 divided by 36 equals 7 rows from one band, and 180 divided by 36 equals 5 rows from the other band.

Compute LCM(10, 15, 25). Using prime factorization: 10 equals 2 multiplied by 5; 15 equals 3 multiplied by 5; 25 equals 5 squared. Taking the highest power of each prime: 2 multiplied by 3 multiplied by 5 squared equals 2 multiplied by 3 multiplied by 25 equals 150. All three events will next occur together after 150 days.

The answer is False. The LCM equals the product only when the two numbers are relatively prime, meaning their GCD equals 1. When GCD is greater than 1, the LCM is less than the product. The correct relationship is LCM(a, b) equals a multiplied by b divided by GCD(a, b). For example, LCM(12, 18) equals 36, but 12 multiplied by 18 equals 216. The product is 6 times larger because GCD(12, 18) equals 6.

The greatest piece length is GCD(126, 84). Using the Euclidean Algorithm: 126 divided by 84 gives remainder 42; 84 divided by 42 gives remainder 0. GCD equals 42 metres. Confirming: 126 divided by 42 equals 3 pieces from the first rope, and 84 divided by 42 equals 2 pieces from the second rope, with no rope left over.

Compute LCM(6, 10, 15). Using prime factorization: 6 equals 2 multiplied by 3; 10 equals 2 multiplied by 5; 15 equals 3 multiplied by 5. Taking the highest power of each prime: 2 multiplied by 3 multiplied by 5 equals 30. All three machines will next complete a cycle together after 30 minutes.