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3. Find the LCM of the following numbers by prime factorization.

a. 36, 40

b. [tex]$15, 30, 50$[/tex]

c. [tex]$12, 30, 90$[/tex]

d. [tex][tex]$64, 80, 100$[/tex][/tex]

e. [tex]$72, 90, 120$[/tex]

f. [tex]$35, 40, 95$[/tex]

Sagot :

GinnyAnsweridn
Sure! Let's calculate the Least Common Multiple (LCM) of each set of numbers using prime factorization.

### a. LCM of 36 and 40
1. Prime factorization:
- 36 = 2² 3²
- 40 = 2³ 5

2. Taking the highest power of each prime:
- [tex]\(2^3, 3^2, 5^1\)[/tex]

3. Multiplying these together:
- [tex]\(LCM = 2^3 3^2 5 = 8 9 5 = 360\)[/tex]

### b. LCM of 15, 30, and 50
1. Prime factorization:
- 15 = 3 5
- 30 = 2 3 5
- 50 = 2 5²

2. Taking the highest power of each prime:
- [tex]\(2^1, 3^1, 5^2\)[/tex]

3. Multiplying these together:
- [tex]\(LCM = 2 3 5^2 = 2 3 25 = 150\)[/tex]

### c. LCM of 12, 30, and 90
1. Prime factorization:
- 12 = 2² 3
- 30 = 2 3 5
- 90 = 2 3² 5

2. Taking the highest power of each prime:
- [tex]\(2^2, 3^2, 5^1\)[/tex]

3. Multiplying these together:
- [tex]\(LCM = 2^2 3^2 5 = 4 9 5 = 180\)[/tex]

### d. LCM of 64, 80, and 100
1. Prime factorization:
- 64 = 2^6
- 80 = 2^4 5
- 100 = 2² 5²

2. Taking the highest power of each prime:
- [tex]\(2^6, 5^2\)[/tex]

3. Multiplying these together:
- [tex]\(LCM = 2^6 5^2 = 64 25 = 1600\)[/tex]

### e. LCM of 72, 90, and 120
1. Prime factorization:
- 72 = 2^3 3²
- 90 = 2 3² 5
- 120 = 2³ 3 5

2. Taking the highest power of each prime:
- [tex]\(2^3, 3^2, 5^1\)[/tex]

3. Multiplying these together:
- [tex]\(LCM = 2^3 3^2 5 = 8 9 5 = 360\)[/tex]

### f. LCM of 35, 40, and 95
1. Prime factorization:
- 35 = 5 7
- 40 = 2³ 5
- 95 = 5 19

2. Taking the highest power of each prime:
- [tex]\(2^3, 5^1, 7^1, 19^1\)[/tex]

3. Multiplying these together:
- [tex]\(LCM = 2^3 5 7 19 = 8 5 7 * 19 = 5320\)[/tex]

So, the LCM values are:
- a. 360
- b. 150
- c. 180
- d. 1600
- e. 360
- f. 5320
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