Question

If the HCF of 210 and 55 is expressed as \(210 \times 5 + 55m\), then find the value of \(m\).

Hint:

To quickly find the HCF of small numbers, you can also use prime factorization: \(210 = 2 \times 3 \times 5 \times 7\) and \(55 = 5 \times 11\). The common factor is clearly 5.

Answer 1

Step 1: Understanding the Concept:
The Highest Common Factor (HCF) can be found using the Prime Factorization method or Euclid's Division Lemma.
Once found, it can be equated to the given linear combination to solve for the unknown variable.
Step 2: Key Formula or Approach:
Euclid's Division Lemma: \(a = bq + r\).
Linear equation: \(HCF(210, 55) = 210 \times 5 + 55m\).
Step 3: Detailed Explanation:
First, find the HCF of 210 and 55:
Using Euclid's algorithm:
\[ 210 = 55 \times 3 + 45 \]
\[ 55 = 45 \times 1 + 10 \]
\[ 45 = 10 \times 4 + 5 \]
\[ 10 = 5 \times 2 + 0 \]
The remainder has become zero, so the HCF is 5.
Now, substitute this value into the given equation:
\[ 5 = 210 \times 5 + 55m \]
\[ 5 = 1050 + 55m \]
Rearrange to solve for \(m\):
\[ 55m = 5 - 1050 \]
\[ 55m = -1045 \]
\[ m = \frac{-1045}{55} \]
Dividing numerator and denominator by 11:
\[ m = \frac{-95}{5} = -19 \]
Step 4: Final Answer:
The value of \(m\) is \(-19\).