Find the HCF of the following :
50 ab and 60 bc

Answer

To find the HCF (Highest Common Factor) of the expressions 50ab and 60bc, we will follow these steps:

### Step 1: Factor the coefficients
First, we will factor the numerical coefficients of both terms.

- For 50ab:
- The prime factorization of 50 is:
\[ 50 = 2 \times 5^2 \]
- Therefore, we can write:
\[ 50ab = 2 \times 5^2 \times a \times b \]

- For 60bc:
- The prime factorization of 60 is:
\[ 60 = 2^2 \times 3 \times 5 \]
- Therefore, we can write:
\[ 60bc = 2^2 \times 3 \times 5 \times b \times c \]

### Step 2: Identify common factors
Next, we will identify the common factors in both expressions.

- From 50ab, we have:
- Factors: \(2, 5, a, b\)

- From 60bc, we have:
- Factors: \(2^2, 3, 5, b, c\)

### Step 3: Determine the HCF of the coefficients
Now, we will find the HCF of the numerical coefficients (50 and 60).

- The common prime factors are:
- \(2\) (minimum power is \(1\))
- \(5\) (minimum power is \(1\))

Thus, the HCF of the coefficients is:
\[ HCF(50, 60) = 2^1 \times 5^1 = 10 \]

### Step 4: Identify common variables
Next, we will look at the variables in both expressions.

- From 50ab, we have \(a\) and \(b\).
- From 60bc, we have \(b\) and \(c\).

The common variable is \(b\).

### Step 5: Combine the HCF of coefficients and common variables
Now, we combine the HCF of the coefficients and the common variable to get the overall HCF.

\[ HCF(50ab, 60bc) = HCF \text{ of coefficients} \times \text{common variables} = 10 \times b = 10b \]

### Final Answer
Thus, the HCF of 50ab and 60bc is:
\[ \boxed{10b} \]

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