Answered

At what time will the bells ring together again?

Given:
[tex]\[
\begin{array}{c|ccc}
2 & 10, & 20, & 30 \\
\hline
5, & 10, & 15 \\
\hline
1, & 2, & 3 \\
\end{array}
\][/tex]

[tex]\[
\begin{aligned}
\therefore \quad \text{L.C.M.} & = 2 \times 5 \times 1 \\
& = 60
\end{aligned}
\][/tex]

The bells will ring together again after 60 minutes. Hence, they will ring at 7 a.m.

EXERCISE 2.4

Find the L.C.M. of the following numbers by prime factorization:

a. [tex]\(20, 24, 30\)[/tex]

b. [tex]\(36, 72, 144\)[/tex]

c. [tex]\(72, 90, 120\)[/tex]

d. [tex]\(40, 96, 120\)[/tex]



Answer :

GinnyAnswer
Certainly! Let's find the Least Common Multiple (LCM) for each set of numbers step-by-step.

### a. Find the LCM of 20, 24, and 30:

1. Prime Factorization:
- 20 = 2^2 5
- 24 = 2^3 3
- 30 = 2 3 5

2. Identify the highest power of each prime:
- 2^3 (highest power of 2)
- 3^1 (highest power of 3)
- 5^1 (highest power of 5)

3. Multiply these together:
- LCM = 2^3 3^1 5^1
- LCM = 8 3 5
- LCM = 24 5
- LCM = 120

So, the LCM of 20, 24, and 30 is 120.

### b. Find the LCM of 36, 72, and 144:

1. Prime Factorization:
- 36 = 2^2 3^2
- 72 = 2^3 3^2
- 144 = 2^4 3^2

2. Identify the highest power of each prime:
- 2^4 (highest power of 2)
- 3^2 (highest power of 3)

3. Multiply these together:
- LCM = 2^4 3^2
- LCM = 16 9
- LCM = 144

So, the LCM of 36, 72, and 144 is 144.

### d. Find the LCM of 72, 90, and 120:

1. Prime Factorization:
- 72 = 2^3 3^2
- 90 = 2^1 3^2 5
- 120 = 2^3 3 5

2. Identify the highest power of each prime:
- 2^3 (highest power of 2)
- 3^2 (highest power of 3)
- 5^1 (highest power of 5)

3. Multiply these together:
- LCM = 2^3 3^2 5
- LCM = 8 9 5
- LCM = 72 5
- LCM = 360

So, the LCM of 72, 90, and 120 is 360.

### e. Find the LCM of 40, 96, and 120:

1. Prime Factorization:
- 40 = 2^3 5
- 96 = 2^5 3
- 120 = 2^3 3 5

2. Identify the highest power of each prime:
- 2^5 (highest power of 2)
- 3^1 (highest power of 3)
- 5^1 (highest power of 5)

3. Multiply these together:
- LCM = 2^5 3 5
- LCM = 32 3 5
- LCM = 96 * 5
- LCM = 480

So, the LCM of 40, 96, and 120 is 480.

### Summary:

a. LCM of 20, 24, and 30 is 120.

b. LCM of 36, 72, and 144 is 144.

d. LCM of 72, 90, and 120 is 360.

e. LCM of 40, 96, and 120 is 480.

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