The Least Common Multiple (LCM) of two numbers is the smallest number that is a multiple of both. For 1200 and 800, the LCM is **4,800**.

### How to Find the LCM

There are different methods to find the LCM of two numbers. Let’s explore the two most common ones:

### 1. **Prime Factorization Method**

In this method, we find the prime factors of each number, then take the highest power of each prime that appears in either factorization.

- Prime factors of 1200:

1200=24×3×521200 = 2^4 \times 3 \times 5^21200=24×3×52 - Prime factors of 800:

800=25×52800 = 2^5 \times 5^2800=25×52

Now, take the highest powers of each prime:

- For 222, the highest power is 252^525
- For 333, we take 313^131 (appears only in 1200)
- For 555, the highest power is 525^252

Multiply them together:

LCM=25×31×52=4800LCM = 2^5 \times 3^1 \times 5^2 = 4800LCM=25×31×52=4800

### 2. **Division Method**

In the division method, we divide the numbers by their common prime factors until we reach 1.

- Step 1: Divide both numbers by 2.

1200÷2=6001200 \div 2 = 6001200÷2=600, 800÷2=400800 \div 2 = 400800÷2=400 - Step 2: Divide again by 2.

600÷2=300600 \div 2 = 300600÷2=300, 400÷2=200400 \div 2 = 200400÷2=200 - Step 3: Divide again by 2.

300÷2=150300 \div 2 = 150300÷2=150, 200÷2=100200 \div 2 = 100200÷2=100 - Step 4: Divide again by 2.

150÷2=75150 \div 2 = 75150÷2=75, 100÷2=50100 \div 2 = 50100÷2=50 - Step 5: Divide both numbers by 3 (since 75 is divisible by 3).

75÷3=2575 \div 3 = 2575÷3=25, 505050 remains the same. - Step 6: Divide both numbers by 5.

25÷5=525 \div 5 = 525÷5=5, 50÷5=1050 \div 5 = 1050÷5=10 - Step 7: Divide again by 5.

5÷5=15 \div 5 = 15÷5=1, 10÷5=210 \div 5 = 210÷5=2 - Step 8: Divide by 2.

2÷2=12 \div 2 = 12÷2=1

Now, multiply all divisors:

LCM=25×3×52=4800LCM = 2^5 \times 3 \times 5^2 = 4800LCM=25×3×52=4800

### Conclusion

The LCM of 1200 and 800 is **4,800**, and you can find it using either the prime factorization method or the division method. Both give the same result