The product of two positive integers is 3026 and their Least Common Multiple (LCM) is 89. Using the relationship between HCF and LCM, find the Highest Common Factor (HCF) of the two numbers.

Introduction / Context:
This question directly tests knowledge of the fundamental relationship between two numbers, their Highest Common Factor (HCF), and their Least Common Multiple (LCM). Many competitive exams include problems where either the product, HCF, or LCM is missing, and you are expected to use a standard formula to find the unknown term. Mastering this relationship makes it easier to handle a large variety of number system questions quickly and accurately.

Given Data / Assumptions:

• Product of the two numbers = 3026.
• LCM of the two numbers = 89.
• The numbers are positive integers.
• We need to find the HCF of the two numbers.

Concept / Approach:
For any two positive integers a and b, there is a well known relationship: a * b = HCF(a, b) * LCM(a, b). This formula connects the product of the numbers with their HCF and LCM. When any two of these three quantities are known, the third can be computed easily. In this problem, we know the product and the LCM, so we can rearrange the formula to find the HCF as: HCF = (product of the numbers) / (LCM).

Step-by-Step Solution:
Step 1: Write the formula. a * b = HCF * LCM Step 2: Substitute the known values. 3026 = HCF * 89 Step 3: Solve for HCF. HCF = 3026 / 89 Step 4: Perform the division. 3026 / 89 = 34 Therefore, the HCF of the two numbers is 34.

Verification / Alternative check:
To verify, we can assume that the two numbers are HCF * m and HCF * n, where m and n are co-prime and their product equals the LCM divided by HCF relationships. However, the quickest verification is simply to multiply the found HCF and the given LCM: 34 * 89. This equals 3026, which matches the original product. This confirms that the calculation is correct and that 34 is the correct HCF.

Why Other Options Are Wrong:
33: If HCF were 33, then HCF * LCM = 33 * 89 = 2937, which does not equal 3026.
35: 35 * 89 = 3115, which does not match the given product.
29: 29 * 89 = 2581, again not equal to 3026.
17: 17 * 89 = 1513, which is exactly half of 3026, so it cannot be the HCF when using the given data.

Common Pitfalls:
Learners sometimes try to factorize the product directly into two numbers and then find their HCF and LCM manually. While possible, that approach is slower and more error prone in a timed exam. Another common mistake is to invert the formula or confuse HCF with LCM. Always remember the clean relationship: product of the numbers equals HCF multiplied by LCM. Using this formula avoids unnecessary effort and leads directly to the answer.

Final Answer:
The HCF of the two numbers is 34.