5. find the prime factorizations of each number. then find the prime factorization of the lcm and calculate…

Explanation:

Step1: Prime - factorize the numbers

a)

For (35 = 5\times7) and (84=2^{2}\times3\times7).

b)

For (15 = 3\times5) and (21 = 3\times7).

c)

For (50=2\times5^{2}) and (60 = 2^{2}\times3\times5).

d)

For (42=2\times3\times7) and (72=2^{3}\times3^{2}).

e)

For (24 = 2^{3}\times3) and (48=2^{4}\times3).

Step2: Find the LCM using prime - factorizations

a)

The LCM of (35) and (84): Take the highest power of each prime factor. For prime factor (2), the highest power is (2^{2}), for (3) it is (3^{1}), for (5) it is (5^{1}) and for (7) it is (7^{1}). So (LCM(35,84)=2^{2}\times3\times5\times7 = 420).

b)

The LCM of (15) and (21): For prime factor (3), the highest power is (3^{1}), for (5) it is (5^{1}) and for (7) it is (7^{1}). So (LCM(15,21)=3\times5\times7 = 105).

c)

The LCM of (50) and (60): For prime factor (2), the highest power is (2^{2}), for (3) it is (3^{1}) and for (5) it is (5^{2}). So (LCM(50,60)=2^{2}\times3\times5^{2}=300).

d)

The LCM of (42) and (72): For prime factor (2), the highest power is (2^{3}), for (3) it is (3^{2}) and for (7) it is (7^{1}). So (LCM(42,72)=2^{3}\times3^{2}\times7 = 504).

e)

The LCM of (24) and (48): For prime factor (2), the highest power is (2^{4}) and for (3) it is (3^{1}). So (LCM(24,48)=2^{4}\times3 = 48).

Answer:

a) Prime - factorizations: (35 = 5\times7), (84=2^{2}\times3\times7), (LCM(35,84)=420) b) Prime - factorizations: (15 = 3\times5), (21 = 3\times7), (LCM(15,21)=105) c) Prime - factorizations: (50=2\times5^{2}), (60 = 2^{2}\times3\times5), (LCM(50,60)=300) d) Prime - factorizations: (42=2\times3\times7), (72=2^{3}\times3^{2}), (LCM(42,72)=504) e) Prime - factorizations: (24 = 2^{3}\times3), (48=2^{4}\times3), (LCM(24,48)=48)