Prime factors and rational numbers

What is the highest common factor (HCF) of 48 and 180?

• 6

• 12

• 24

• 36

Answer explanation

To find the HCF of 48 and 180, we can list the factors: Factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48; Factors of 180 are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180. The highest common factor is 12.

Find the lowest common multiple (LCM) of 14 and 35.

• 70

• 140

• 210

• 490

Answer explanation

To find the LCM of 14 and 35, we can use the prime factorization method. The prime factors of 14 are 2 and 7, and for 35, they are 5 and 7. The LCM is found by taking the highest power of each prime: 2^1, 5^1, and 7^1. Thus, LCM = 2 * 5 * 7 = 70.

Using a factor tree, determine the prime factorization of 84.

Answer explanation

To factor 84, start with 84 = 2 × 42. Then, 42 = 2 × 21, and 21 = 3 × 7. Thus, the prime factorization is 2^2 × 3 × 7, confirming the correct choice is 2^2 × 3 × 7.

A gardener has 60 roses and 84 tulips. She wants to plant them in rows such that each row has the same number of flowers and only one type of flower. What is the maximum number of flowers she can plant in each row?

• 6

• 12

• 24

• 28

Answer explanation

To find the maximum number of flowers per row, we need the greatest common divisor (GCD) of 60 and 84. The GCD is 12, meaning the gardener can plant 12 flowers in each row, maximizing the number of rows.

Two buses leave a station at the same time. One bus returns every 45 minutes, and the other every 60 minutes. After how many minutes will both buses return to the station at the same time?

• 90

• 120

• 180

• 240

Answer explanation

To find when both buses return together, calculate the least common multiple (LCM) of 45 and 60. The LCM is 180 minutes, meaning both buses will return to the station at the same time after 180 minutes.

What is the highest common factor of 56, 72, and 96?

• 4

• 8

• 12

• 16

Answer explanation

To find the highest common factor (HCF) of 56, 72, and 96, we can list their factors. The common factors are 1, 2, 4, and 8. The highest of these is 8, making it the correct answer.

Find the lowest common multiple of 9, 12, and 15.

• 90

• 180

• 360

• 540

Answer explanation

To find the LCM of 9, 12, and 15, we determine the prime factors: 9 = 3^2, 12 = 2^2 * 3, 15 = 3 * 5. The LCM is found by taking the highest power of each prime: 2^2 * 3^2 * 5 = 360. Thus, the correct answer is 360.

Using a factor tree, determine the prime factorization of 210.

Answer explanation

To find the prime factorization of 210, we can use a factor tree. 210 can be divided by 2 (210 = 2 × 105), then 105 can be divided by 3 (105 = 3 × 35), and finally, 35 can be divided by 5 and 7 (35 = 5 × 7). Thus, the prime factorization is 2 × 3 × 5 × 7.

A teacher has 48 pencils and 64 erasers. She wants to distribute them equally among her students such that each student gets the same number of pencils and erasers. What is the maximum number of students she can distribute them to?

• 4

• 8

• 12

• 16

Answer explanation

To find the maximum number of students, we need the greatest common divisor (GCD) of 48 and 64. The GCD is 16, meaning the teacher can distribute the pencils and erasers equally among 16 students.

Two gears are rotating together. One gear completes a full rotation every 18 seconds, and the other every 24 seconds. After how many seconds will both gears align again at the starting position?

• 36

• 72

• 108

• 144

Answer explanation

To find when both gears align, calculate the least common multiple (LCM) of their rotation times: LCM(18, 24) = 72 seconds. Thus, both gears will align again at the starting position after 72 seconds.