Difficulty: Easy
Correct Answer: 865
Explanation:
Introduction / Context:
This question asks for the highest common factor (HCF) of two numbers, 865 and 2595. The HCF is widely used in simplifying fractions, solving ratio problems, and understanding divisibility properties. When one number is a factor of another, the HCF turns out to be the smaller number, which is an important observation in such problems.
Given Data / Assumptions:
Concept / Approach:
If one number divides the other exactly, the smaller number is the HCF. To check this, we can perform division or use factorization. Alternatively, we can apply the Euclidean algorithm, which repeatedly uses the remainder to reduce the problem until the remainder becomes zero. The last non zero remainder is the HCF.
Step-by-Step Solution:
Observe that 2595 = 3 * 865.
Compute: 865 * 3 = 2595.
This means 865 divides 2595 exactly with no remainder.
Also, of course, 865 divides itself exactly.
Therefore, the HCF of 865 and 2595 is 865.
Verification / Alternative check:
Using the Euclidean algorithm:
2595 divided by 865 gives quotient 3 and remainder 0.
When the remainder becomes 0, the divisor at that step is the HCF.
Thus, HCF(2595, 865) = 865.
This confirms our earlier reasoning that 865 is the highest common factor.
Why Other Options Are Wrong:
5, 25, 35, and 95 are all factors of 865 and 2595, but they are smaller than 865. Since 865 itself divides both numbers exactly, it is clearly the greatest such factor, and the others cannot be the HCF.
Common Pitfalls:
Students sometimes stop after finding a small common factor and forget to check if one number is a factor of the other. Another error is to confuse HCF with LCM and attempt unnecessary multiplications instead of looking for the largest common divisor. Always check whether the smaller number divides the larger one before moving to more complicated methods.
Final Answer:
865
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