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: Find HCF of 210 and 55
Using Euclid’s Division Algorithm:

210 = 55 × 3 + 45
55 = 45 × 1 + 10
45 = 10 × 4 + 5
10 = 5 × 2 + 0

Since remainder becomes 0,
HCF(210, 55) = 5

Step 2: Substitute in Given Equation
Given:
HCF = 210 × 5 + 55m

So,
5 = 210 × 5 + 55m
5 = 1050 + 55m

Step 3: Solve for m
55m = 5 − 1050
55m = −1045

m = −1045 / 55

Divide by 11:
= −95 / 5
= −19

Final Answer:
m = −19