If the HCF (GCD) of 210 and 55 can be expressed in the form: HCF(210, 55) = 210 × 5 + 55 × P then what is the value of P?

Introduction / Context:
This question checks your understanding of Bézout's identity and the Euclidean algorithm. The HCF of two integers can always be written as a linear combination of those integers. Here, the HCF of 210 and 55 is expressed as 210 * 5 + 55 * P, and we need to determine P.

Given Data / Assumptions:

• Numbers: 210 and 55
• HCF(210, 55) is expressed as 210 * 5 + 55 * P
• We need to find P

Concept / Approach:
First, compute the HCF of 210 and 55 using the Euclidean algorithm. Then set that value equal to the expression 210 * 5 + 55 * P and solve for P. Note that P can be negative, which is completely acceptable in a linear combination representation.

Step-by-Step Solution:
Step 1: Find gcd(210, 55).210 mod 55 = 45.55 mod 45 = 10.45 mod 10 = 5.10 mod 5 = 0, so gcd(210, 55) = 5.Step 2: Use the given expression:HCF(210, 55) = 210 * 5 + 55 * P.So 5 = 210 * 5 + 55 * P.Step 3: Compute 210 * 5 = 1050.Thus 5 = 1050 + 55P.Step 4: Rearrange to solve for P: 55P = 5 - 1050 = -1045.P = -1045 / 55 = -19.

Verification / Alternative check:
Substitute P = -19 back into the expression: 210 * 5 + 55 * (-19) = 1050 - 1045 = 5, which matches the HCF of 210 and 55. This confirms that P = -19 is correct.

Why Other Options Are Wrong:
-23: 210 * 5 + 55 * (-23) = 1050 - 1265 = -215, not 5.16: 210 * 5 + 55 * 16 = 1050 + 880 = 1930, not 5.27: 210 * 5 + 55 * 27 = 1050 + 1485 = 2535, not 5.-9: 210 * 5 + 55 * (-9) = 1050 - 495 = 555, not 5.

Common Pitfalls:
Failing to compute the HCF correctly with the Euclidean algorithm.Forgetting that P can be negative, which is allowed in linear combinations.Making sign errors when rearranging the equation 5 = 1050 + 55P.

Final Answer:
-19