Difficulty: Medium
Correct Answer: -19
Explanation:
Introduction:
This question tests your understanding of the Euclidean algorithm and the idea that the highest common factor (H.C.F.) of two integers can be expressed as a linear combination of those integers. It is a fundamental concept in number theory and is often used to solve Diophantine equations.
Given Data / Assumptions:
Concept / Approach:
First, we compute the H.C.F. of 210 and 55 using either prime factorisation or the Euclidean algorithm. Then we equate this H.C.F. to the given expression 210 × 5 + 55P and solve for P. Note that the linear combination may involve a negative coefficient.
Step-by-Step Solution:
Step 1: Compute HCF(210, 55). 210 ÷ 55 = 3 remainder 45 (since 55 * 3 = 165 and 210 - 165 = 45). 55 ÷ 45 = 1 remainder 10. 45 ÷ 10 = 4 remainder 5. 10 ÷ 5 = 2 remainder 0. Therefore, H.C.F. = 5. Step 2: Given that HCF = 210 × 5 + 55P. Step 3: Substitute HCF = 5 into the expression: 5 = 210 × 5 + 55P. Step 4: Compute 210 × 5 = 1050. Step 5: So 5 = 1050 + 55P. Step 6: Rearrange: 55P = 5 - 1050 = -1045. Step 7: Therefore, P = -1045 / 55 = -19.
Verification / Alternative check:
Verify that 210 × 5 + 55 × (-19) = 1050 - 1045 = 5, which matches the H.C.F. This confirms that P = -19 is correct and that the linear combination indeed yields the greatest common divisor.
Why Other Options Are Wrong:
-23: Would give 210 × 5 + 55 × (-23) = 1050 - 1265 = -215, not equal to 5. 16 and 27: These positive values of P make 210 × 5 + 55P far larger than 5 and not equal to the H.C.F. 5: Substituting gives 210 × 5 + 55 × 5 = 1050 + 275 = 1325, again not the H.C.F.
Common Pitfalls:
Students sometimes assume that both coefficients in the linear combination must be positive, which is not true. Others may miscalculate the H.C.F. or make algebraic errors while solving for P. Carefully following the steps of the Euclidean algorithm and algebraic manipulation avoids these mistakes.
Final Answer:
The correct value of P is -19.
Discussion & Comments