Finding the Least Common Multiple (LCM) of 56 and 50

The Least Common Multiple (LCM) of two or more numbers is the smallest positive integer that is a multiple of all the given numbers. It's often used when working with fractions or in problems involving cycles or repetitions.

There are several ways to find the LCM, but the prime factorization method is very common and reliable. Let's find the prime factorization of 56 and 50.

Prime Factorization of 56

We break down 56 into its prime factors:

• 56 is divisible by 2: \(56 \div 2 = 28\)
• 28 is divisible by 2: \(28 \div 2 = 14\)
• 14 is divisible by 2: \(14 \div 2 = 7\)
• 7 is a prime number.

So, the prime factorization of 56 is \(2 \times 2 \times 2 \times 7\), which can be written in exponential form as \(2^3 \times 7^1\).

Prime Factorization of 50

Now, let's break down 50 into its prime factors:

• 50 is divisible by 2: \(50 \div 2 = 25\)
• 25 is divisible by 5: \(25 \div 5 = 5\)
• 5 is a prime number.

So, the prime factorization of 50 is \(2 \times 5 \times 5\), which can be written in exponential form as \(2^1 \times 5^2\).

Calculating the LCM using Prime Factors

To find the LCM of 56 and 50, we take the highest power of each prime factor that appears in either factorization. The prime factors involved are 2, 5, and 7.

• For the prime factor 2: The powers are \(2^3\) (from 56) and \(2^1\) (from 50). The highest power is \(2^3\).
• For the prime factor 5: The powers are \(5^0\) (effectively, as 5 does not appear in 56) and \(5^2\) (from 50). The highest power is \(5^2\).
• For the prime factor 7: The powers are \(7^1\) (from 56) and \(7^0\) (effectively, as 7 does not appear in 50). The highest power is \(7^1\).

Now, we multiply these highest powers together to get the LCM:

LCM(56, 50) = \(2^3 \times 5^2 \times 7^1 = 8 \times 25 \times 7\)

Let's calculate the product:

\(8 \times 25 = 200\)

\(200 \times 7 = 1400\)

So, the Least Common Multiple of 56 and 50 is 1400.

Checking the options, we see that 1400 is one of the choices.

Revision Table: Key Concepts in LCM Calculation

Additional Information on Finding LCM and HCF

The LCM is closely related to the Highest Common Factor (HCF) or Greatest Common Divisor (GCD). The HCF is the largest positive integer that divides two or more numbers without leaving a remainder.

There's a useful relationship between the LCM and HCF of two numbers, say 'a' and 'b':

\(\text{LCM}(a, b) \times \text{HCF}(a, b) = a \times b\)

Let's quickly find the HCF of 56 and 50 using their prime factorizations:

\(56 = 2^3 \times 7^1\)

\(50 = 2^1 \times 5^2\)

To find the HCF, we take the lowest power of each common prime factor. The only common prime factor is 2, and the lowest power is \(2^1\).

HCF(56, 50) = \(2^1 = 2\)

Now, let's check the relationship:

LCM(56, 50) \(\times\) HCF(56, 50) = \(1400 \times 2 = 2800\)

\(a \times b = 56 \times 50\)

\(56 \times 50 = 56 \times 5 \times 10 = 280 \times 10 = 2800\)

The relationship holds true: \(1400 \times 2 = 56 \times 50\), which is \(2800 = 2800\). This confirms our calculation of the LCM is likely correct.