Question

1. determine the least common multiple of each set of numbers.

a) 12, 15, 21
b) 12, 20, 32
c) 18, 24, 30
d) 30, 32, 40
e) 49, 56, 64
f) 50, 55, 66

Explanation:

Step1: Prime factorize each number

Part a)

$12=2^2\times3$, $15=3\times5$, $21=3\times7$

Step2: Take max exponents of primes

$\text{LCM}=2^2\times3\times5\times7$
$\text{LCM}=4\times3\times5\times7=420$

---

Step1: Prime factorize each number

Part b)

$12=2^2\times3$, $20=2^2\times5$, $32=2^5$

Step2: Take max exponents of primes

$\text{LCM}=2^5\times3\times5$
$\text{LCM}=32\times3\times5=480$

---

Step1: Prime factorize each number

Part c)

$18=2\times3^2$, $24=2^3\times3$, $30=2\times3\times5$

Step2: Take max exponents of primes

$\text{LCM}=2^3\times3^2\times5$
$\text{LCM}=8\times9\times5=360$

---

Step1: Prime factorize each number

Part d)

$30=2\times3\times5$, $32=2^5$, $40=2^3\times5$

Step2: Take max exponents of primes

$\text{LCM}=2^5\times3\times5$
$\text{LCM}=32\times3\times5=480$

---

Step1: Prime factorize each number

Part e)

$49=7^2$, $56=2^3\times7$, $64=2^6$

Step2: Take max exponents of primes

$\text{LCM}=2^6\times7^2$
$\text{LCM}=64\times49=3136$

---

Step1: Prime factorize each number

Part f)

$50=2\times5^2$, $55=5\times11$, $66=2\times3\times11$

Step2: Take max exponents of primes

$\text{LCM}=2\times3\times5^2\times11$
$\text{LCM}=2\times3\times25\times11=1650$

Answer:

a) 420
b) 480
c) 360
d) 480
e) 3136
f) 1650