Question

learning goal 2.1: students will find the greatest common factor (gcf) and rewrite fractions in simplest form
find the greatest common factor (gcf) of each pair of numbers

1. 36 and 54
2. 17 and 91
3. 64 and 315
4. 374 and 622

learning goal 2.2: students will find the least common multiple (lcm) in mathematical and real world contexts.
find the least common multiple (lcm) of each pair of numbers

1. 12 and 15
2. 7 and 11
3. 35 and 5
4. 24 and 16

Explanation:

Step1: Find LCM of 12 and 15

Factorize 12: $12 = 2^2 \times 3$
Factorize 15: $15 = 3 \times 5$
LCM is product of highest powers: $2^2 \times 3 \times 5 = 60$

Step2: Find LCM of 7 and 11

7 and 11 are primes. LCM(7,11) = $7 \times 11 = 77$

Step3: Find LCM of 35 and 5

Factorize 35: $35 = 5 \times 7$
Factorize 5: $5 = 5$
LCM is 35 (since 35 is multiple of 5)

Step4: Find LCM of 24 and 16

Factorize 24: $24 = 2^3 \times 3$
Factorize 16: $16 = 2^4$
LCM is $2^4 \times 3 = 48$

Answer:

1. LCM of 12 and 15: $\boldsymbol{60}$
2. LCM of 7 and 11: $\boldsymbol{77}$
3. LCM of 35 and 5: $\boldsymbol{35}$
4. LCM of 24 and 16: $\boldsymbol{48}$