GCD Master Quiz: Greatest Common Divisor Challenge

Using prime factorization: 105 equals 3 multiplied by 5 multiplied by 7, and 70 equals 2 multiplied by 5 multiplied by 7. The common prime factors are 5 and 7. Multiplying them gives 5 multiplied by 7 equals 35. So GCD(105, 70) = 35. This means 35 is the largest number that divides both 105 and 70 exactly: 105 divided by 35 equals 3, and 70 divided by 35 equals 2, with no remainder.

Using prime factorization: 132 equals 2 squared multiplied by 3 multiplied by 11, and 88 equals 2 cubed multiplied by 11. The common prime factors are 2 squared and 11. Multiplying them gives 4 multiplied by 11 equals 44. So GCD(132, 88) = 44. Confirming: 132 divided by 44 equals 3, and 88 divided by 44 equals 2, with no remainder in either case.

Prime factorization: 210 equals 2 multiplied by 3 multiplied by 5 multiplied by 7, and 126 equals 2 multiplied by 3 squared multiplied by 7. The common prime factors are 2, 3, and 7. Multiplying them gives 2 multiplied by 3 multiplied by 7 equals 42. So GCD(210, 126) = 42. Confirming: 210 divided by 42 equals 5, and 126 divided by 42 equals 3, with no remainder.

The answer is True. To find the GCD using prime factorization, each number is broken down into its prime factors. Any prime factor that appears in both factorizations is a common factor. When all common prime factors are multiplied together, taking each common prime factor the smaller number of times it appears in either number, the result is the GCD. For example, GCD(36, 48): 36 equals 2 squared multiplied by 3 squared and 48 equals 2 to the fourth multiplied by 3. Common factors are 2 squared and 3, giving GCD = 12.

Using prime factorization: 196 equals 2 squared multiplied by 7 squared, and 84 equals 2 squared multiplied by 3 multiplied by 7. The common prime factors are 2 squared and 7. Multiplying them gives 4 multiplied by 7 equals 28. So GCD(196, 84) = 28. Confirming: 196 divided by 28 equals 7, and 84 divided by 28 equals 3, with no remainder in either case.

Option A is correct: 2 multiplied by 3 multiplied by 5 multiplied by 5 equals 150. Option B is correct: 3 multiplied by 3 multiplied by 5 multiplied by 5 equals 225. Option C is correct: the common prime factors of 150 and 225 are 3 and 5 squared, since 2 appears only in 150 and not in 225. Multiplying 3 multiplied by 5 multiplied by 5 gives 75, so GCD = 75. Option D is wrong because it incorrectly includes 2 as a common factor and omits one factor of 5, giving 30 instead of 75.

Using prime factorization: 280 equals 2 cubed multiplied by 5 multiplied by 7, and 168 equals 2 cubed multiplied by 3 multiplied by 7. The common prime factors are 2 cubed and 7. Multiplying them gives 8 multiplied by 7 equals 56. So GCD(280, 168) = 56. Confirming: 280 divided by 56 equals 5, and 168 divided by 56 equals 3, with no remainder in either case.

Using prime factorization: 112 equals 2 to the fourth multiplied by 7, and 70 equals 2 multiplied by 5 multiplied by 7. The common prime factors are 2 and 7. Multiplying them gives 2 multiplied by 7 equals 14. So GCD(112, 70) = 14. Confirming: 112 divided by 14 equals 8, and 70 divided by 14 equals 5, with no remainder in either case.

The answer is False. Neither method is always faster than the other for every pair of numbers. The Euclidean Algorithm can be more efficient for large numbers because it avoids finding complete prime factorizations, which becomes time-consuming when numbers have large prime factors. Prime factorization can be quicker when numbers have small, obvious prime factors. The most efficient method depends on the specific numbers involved.

Since 126 divided by 42 equals exactly 3 with no remainder, 42 divides 126 exactly. When one number divides the other exactly, the GCD equals the smaller number. Therefore GCD(126, 42) = 42. This can also be confirmed by prime factorization: 126 equals 2 multiplied by 3 squared multiplied by 7, and 42 equals 2 multiplied by 3 multiplied by 7. The common factors are 2 multiplied by 3 multiplied by 7 equals 42.

The greatest piece length that divides both 63 and 105 exactly is GCD(63, 105). Using prime factorization: 63 equals 3 squared multiplied by 7, and 105 equals 3 multiplied by 5 multiplied by 7. The common prime factors are 3 and 7. Multiplying them gives 3 multiplied by 7 equals 21. So GCD(63, 105) = 21 cm. Confirming: 63 divided by 21 equals 3 pieces per red ribbon, and 105 divided by 21 equals 5 pieces per blue ribbon.

Using prime factorization: 180 equals 2 squared multiplied by 3 squared multiplied by 5, and 135 equals 3 cubed multiplied by 5. The common prime factors are 3 squared and 5. Multiplying them gives 9 multiplied by 5 equals 45. So GCD(180, 135) = 45. Confirming: 180 divided by 45 equals 4, and 135 divided by 45 equals 3, with no remainder in either case.

GCD(112, 70): 112 = 2 to the fourth multiplied by 7 and 70 = 2 multiplied by 5 multiplied by 7. Common = 2 multiplied by 7 = 14. Correct. GCD(42, 56): 42 = 2 multiplied by 3 multiplied by 7 and 56 = 2 cubed multiplied by 7. Common = 2 multiplied by 7 = 14. Correct. GCD(28, 42): 28 = 2 squared multiplied by 7 and 42 = 2 multiplied by 3 multiplied by 7. Common = 2 multiplied by 7 = 14. Correct. GCD(35, 63): 35 = 5 multiplied by 7 and 63 = 3 squared multiplied by 7. Common = 7 only. GCD = 7, not 14. Option D does not qualify.

Using prime factorization: 150 equals 2 multiplied by 3 multiplied by 5 squared, and 225 equals 3 squared multiplied by 5 squared. The common prime factors are 3 and 5 squared. Multiplying them gives 3 multiplied by 25 equals 75. So GCD(150, 225) = 75. Confirming: 150 divided by 75 equals 2, and 225 divided by 75 equals 3, with no remainder in either case.

To simplify a fraction, divide both the numerator and denominator by their GCD. GCD(132, 88) = 44, since 132 = 2 squared multiplied by 3 multiplied by 11 and 88 = 2 cubed multiplied by 11, giving common factors 2 squared multiplied by 11 = 44. Dividing: 132 divided by 44 equals 3, and 88 divided by 44 equals 2. The simplified fraction is 3 over 2. Option B equals the same value but is not fully simplified since 6 and 4 share a common factor of 2.

The answer is False. Two numbers can have multiple common factors. For example, 36 and 48 share the common factors 1, 2, 3, 4, 6, and 12. The GCD is 12, the greatest of all these common factors. The other common factors such as 2, 3, 4, and 6 all divide both numbers exactly but are smaller than the GCD. The GCD is the largest common factor, not the only one.

Using prime factorization: 144 equals 2 to the fourth multiplied by 3 squared, and 60 equals 2 squared multiplied by 3 multiplied by 5. The common prime factors are 2 squared and 3. Multiplying them gives 4 multiplied by 3 equals 12. So GCD(144, 60) = 12. Confirming: 144 divided by 12 equals 12, and 60 divided by 12 equals 5, with no remainder in either case.

The largest square tile that fits exactly must have a side length equal to GCD(120, 168). Using prime factorization: 120 equals 2 cubed multiplied by 3 multiplied by 5, and 168 equals 2 cubed multiplied by 3 multiplied by 7. The common prime factors are 2 cubed and 3. Multiplying gives 8 multiplied by 3 equals 24. So GCD(120, 168) = 24 cm. Confirming: 120 divided by 24 equals 5 tiles across, and 168 divided by 24 equals 7 tiles down, covering the floor with 35 tiles total.

The answer is True. When a prime factor appears in both numbers, it contributes to the GCD at the lower of its two exponents. If a prime p appears as p squared in one number and p cubed in the other, the smaller exponent is 2, so p squared is included in the GCD. For example, in GCD(36, 48): the factor 2 appears as 2 squared in 36 and 2 to the fourth in 48. The smaller exponent is 2, so 2 squared contributes to the GCD, giving 4 as part of the result.

The greatest number of identical groups is GCD(56, 84). Using prime factorization: 56 equals 2 cubed multiplied by 7, and 84 equals 2 squared multiplied by 3 multiplied by 7. The common prime factors are 2 squared and 7. Multiplying gives 4 multiplied by 7 equals 28. So GCD(56, 84) = 28 groups. Each fiction group contains 56 divided by 28 equals 2 books, and each non-fiction group contains 84 divided by 28 equals 3 books.