Mastering Prime Factorization and LCM

What are the prime factors of 84?

• 2, 3, 7

• 2, 3, 5

• 3, 5, 7

• 2, 5, 7

Answer explanation

To find the prime factors of 84, we can divide it by the smallest prime numbers. 84 = 2 x 42, 42 = 2 x 21, 21 = 3 x 7. Thus, the prime factors are 2, 3, and 7, making the correct choice 2, 3, 7.

Find the highest common factor (HCF) of 48 and 60.

• 6

• 8

• 12

• 24

Answer explanation

To find the HCF of 48 and 60, we can list the factors: 48 (1, 2, 3, 4, 6, 8, 12, 16, 24, 48) and 60 (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60). The highest common factor is 12.

Determine the lowest common multiple (LCM) of 15 and 20.

• 40

• 60

• 80

• 100

Answer explanation

To find the LCM of 15 and 20, list the multiples: 15 (15, 30, 45, 60, ...) and 20 (20, 40, 60, ...). The smallest common multiple is 60, making it the LCM.

Using prime factorization, find the HCF of 36 and 54.

• 6

• 9

• 12

• 18

Answer explanation

The prime factorization of 36 is 2^2 × 3^2 and for 54 is 2 × 3^3. The HCF is found by taking the lowest powers of common prime factors: 2^1 and 3^2, which gives us 2 × 9 = 18.

A gardener is planting flowers in rows. If he has 24 tulips and 36 roses, what is the greatest number of rows he can plant so that each row has the same number of flowers and only one type of flower?

• 6

• 8

• 12

• 18

Answer explanation

To find the greatest number of rows, we need the greatest common divisor (GCD) of 24 and 36. The GCD is 6, meaning the gardener can plant 6 rows with 4 tulips or 6 roses in each row.

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 prime factorization of 90?

Answer explanation

To find the prime factorization of 90, we divide it by the smallest prime numbers: 90 = 2 × 45, 45 = 3 × 15, 15 = 3 × 5. Thus, the prime factorization is 2 × 3^2 × 5, which matches the correct choice.

Find the LCM of 9, 12, and 15.

• 60

• 90

• 180

• 360

Answer explanation

To find the LCM of 9, 12, and 15, we determine the prime factorization: 9 = 3^2, 12 = 2^2 * 3, and 15 = 3 * 5. The LCM is found by taking the highest power of each prime: 2^2 * 3^2 * 5 = 180. Thus, the correct answer is 180.

A teacher has 30 pencils and 45 erasers. She wants to distribute them equally among her students without any remainder. What is the maximum number of students she can distribute them to?

• 5

• 10

• 15

• 20

Answer explanation

To find the maximum number of students, we need the greatest common divisor (GCD) of 30 and 45. The GCD is 15, meaning the teacher can distribute the pencils and erasers equally among 15 students without any remainder.

If the LCM of two numbers is 84 and their HCF is 6, which of the following could be the two numbers?

• 12 and 42

• 14 and 36

• 18 and 28

• 24 and 30

Answer explanation

The LCM of two numbers is equal to the product of the numbers divided by their HCF. For 12 and 42, LCM = (12 * 42) / 6 = 84. The other pairs do not satisfy this condition.