Prime Factorization for GCF and LCM (B) Worksheet | Fun and ... - Free Printable

To solve this worksheet, we need to break down each number into its prime factors. Prime factors are the prime numbers that multiply together to make the original number.

Step 1: Prime Factorization for Section A

1. Number 24:
* Start with the smallest prime number, 2.
* $24 \div 2 = 12$
* $12 \div 2 = 6$
* $6 \div 2 = 3$
* 3 is a prime number.
* So, the factors are $2 \times 2 \times 2 \times 3$.

2. Number 36:
* $36 \div 2 = 18$
* $18 \div 2 = 9$
* 9 is not divisible by 2, so try the next prime, 3.
* $9 \div 3 = 3$
* 3 is prime.
* So, the factors are $2 \times 2 \times 3 \times 3$.

3. Number 60:
* $60 \div 2 = 30$
* $30 \div 2 = 15$
* 15 is not divisible by 2. Try 3.
* $15 \div 3 = 5$
* 5 is prime.
* So, the factors are $2 \times 2 \times 3 \times 5$.

4. Number 98:
* $98 \div 2 = 49$
* 49 is not divisible by 2 or 3. Try 5 (no), then 7.
* $49 \div 7 = 7$
* 7 is prime.
* So, the factors are $2 \times 7 \times 7$.

5. Number 130:
* $130 \div 2 = 65$
* 65 ends in 5, so divide by 5.
* $65 \div 5 = 13$
* 13 is prime.
* So, the factors are $2 \times 5 \times 13$.

6. Number 308:
* $308 \div 2 = 154$
* $154 \div 2 = 77$
* 77 is not divisible by 2 or 3. Try 5 (no), then 7.
* $77 \div 7 = 11$
* 11 is prime.
* So, the factors are $2 \times 2 \times 7 \times 11$.

Step 2: Calculate LCM and GCF

a) The LCM of 24 and 36
* Factors of 24: $2 \times 2 \times 2 \times 3$
* Factors of 36: $2 \times 2 \times 3 \times 3$
* To find the Least Common Multiple (LCM), we take the highest power of every prime factor present.
* Highest power of 2: $2 \times 2 \times 2$ (from 24)
* Highest power of 3: $3 \times 3$ (from 36)
* Calculation: $8 \times 9 = 72$.

b) The LCM of 60 and 98
* Factors of 60: $2 \times 2 \times 3 \times 5$
* Factors of 98: $2 \times 7 \times 7$
* To find the LCM, take the highest power of every prime factor.
* Highest power of 2: $2 \times 2$ (from 60)
* Highest power of 3: 3 (from 60)
* Highest power of 5: 5 (from 60)
* Highest power of 7: $7 \times 7$ (from 98)
* Calculation: $4 \times 3 \times 5 \times 49 = 60 \times 49$.
* $60 \times 40 = 2400$
* $60 \times 9 = 540$
* $2400 + 540 = 2940$.

c) The GCF of 36 and 308
* Factors of 36: $2 \times 2 \times 3 \times 3$
* Factors of 308: $2 \times 2 \times 7 \times 11$
* To find the Greatest Common Factor (GCF), we multiply the prime factors that appear in *both* lists.
* Common factors: $2 \times 2$.
* Calculation: $4$.

d) The GCF of 98 and 308
* Factors of 98: $2 \times 7 \times 7$
* Factors of 308: $2 \times 2 \times 7 \times 11$
* To find the GCF, multiply the common prime factors.
* Common factors: $2 \times 7$.
* Calculation: $14$.

Final Answer:
Section A:
24 = $2 \times 2 \times 2 \times 3$
36 = $2 \times 2 \times 3 \times 3$
60 = $2 \times 2 \times 3 \times 5$
98 = $2 \times 7 \times 7$
130 = $2 \times 5 \times 13$
308 = $2 \times 2 \times 7 \times 11$

Calculations:
a) 72
b) 2,940
c) 4
d) 14