Question

1. find the prime factors. fill in the table. find the lcm and gcf for the pair of numbers. hint: list all factors least to greatest. given: 27, 90 prime factors: 27= 3 ·3 ·3 90= 2 ·3 ·3 ·5 lcm = 270 gcf= 9

Explanation:

Step1: Fill LCM table missing values

For the LCM row, take the highest count of each prime factor:

• Prime 2: take 2 (from 90)
• Prime 5: take 5 (from 90)

So LCM row: $2, 3, 3, 3, 5$

Step2: Verify LCM calculation

Calculate LCM using prime factors:
$$2 \times 3 \times 3 \times 3 \times 5 = 270$$

Step3: Fill GCF table missing values

For the GCF row, take the lowest count of each shared prime factor:

• 27's prime factors: $3, 3, 3$
• 90's prime factors: $2, 3, 3, 5$
• Shared prime: 3 (lowest count is 2)
• Non-shared primes (2,5) are excluded

So 27 row: $3, 3$; 90 row: $2, 5$; GCF row: $3, 3$

Step4: Verify GCF calculation

Calculate GCF using shared primes:
$$3 \times 3 = 9$$

Answer:

Filled Table Values:

LCM Table:

LCM row missing boxes: $2$, $5$

GCF Table:

27 row missing boxes: $3$, $3$
90 row missing boxes: $2$, $5$
GCF row missing boxes: $3$, $3$

Final Results:

LCM = $270$
GCF = $9$