Question

e91b
(8) (28, 42) →
(9) (12, 21) →
(10) (18, 24) →
(11) (15, 25) →
(12) (24, 30) →
note (8) you can find the lcm using the methods shown below.
14)28,42
2, 3
4 × 2 × 3 = 84
one step
7)28,42

1. 4, 6

2, 3
7 × 2 × 2 × 3 = 84
more than one step

Explanation:

Step1: Find LCM of 28 and 42

Prime factorize 28: \(28 = 2^2\times7\)
Prime factorize 42: \(42 = 2\times3\times7\)
LCM is product of highest powers: \(2^2\times3\times7 = 84\)

Step2: Find LCM of 12 and 21

Prime factorize 12: \(12 = 2^2\times3\)
Prime factorize 21: \(21 = 3\times7\)
LCM: \(2^2\times3\times7 = 84\)? Wait, no: \(2^2\times3\times7 = 84\)? Wait, 12 and 21: highest 2², 3¹, 7¹. So \(4\times3\times7 = 84\)? Wait no, 12 and 21: 12=2²×3, 21=3×7. LCM is 2²×3×7=84? Wait no, 12×7=84, 21×4=84. Yes.
Wait, no, let's do it step by step. Using the method from the note. For (12,21):
Find GCD first? Or use prime factors.
Prime factors: 12=2×2×3, 21=3×7. LCM is (12×21)/GCD(12,21). GCD(12,21)=3. So (12×21)/3=84. Wait, but let's check the method from the note. Let's use the division method.
Divide 12 and 21 by 3: 4,7. Then LCM is 3×4×7=84.

Step3: Find LCM of 18 and 24

Prime factorize 18: \(18 = 2\times3^2\)
Prime factorize 24: \(24 = 2^3\times3\)
LCM: \(2^3\times3^2 = 8\times9 = 72\)
Using division method: divide by 2: 9,12; divide by 3: 3,4. LCM: 2×3×3×4=72.

Step4: Find LCM of 15 and 25

Prime factorize 15: \(15 = 3\times5\)
Prime factorize 25: \(25 = 5^2\)
LCM: \(3\times5^2 = 75\)
Using division: divide by 5: 3,5. LCM: 5×3×5=75.

Step5: Find LCM of 24 and 30

Prime factorize 24: \(24 = 2^3\times3\)
Prime factorize 30: \(30 = 2\times3\times5\)
LCM: \(2^3\times3\times5 = 120\)
Using division: divide by 2: 12,15; divide by 3: 4,5. LCM: 2×3×4×5=120.

Answer:

(8) 84
(9) 84
(10) 72
(11) 75
(12) 120