What is the Highest Common Factor (HCF) of the two numbers 133 and 112?

Introduction / Context:
This question asks for the Highest Common Factor, or HCF, of two numbers, 133 and 112. HCF problems are fundamental in arithmetic and appear frequently in exams. Understanding how to find the largest integer that divides both given numbers exactly helps in simplifying fractions, solving ratio problems, and dealing with many forms of divisibility based reasoning questions.

Given Data / Assumptions:

• First number = 133.
• Second number = 112.
• We need to determine their HCF.
• Both numbers are positive integers.

Concept / Approach:
We can find the HCF using prime factorization or the Euclidean algorithm. For moderate size numbers like these, it is often very convenient to factor them into primes. Once factorizations are obtained, the HCF is the product of all prime factors common to both numbers, taken with the lowest power that appears in either factorization. Alternatively, we can keep dividing by common primes step by step until no further common prime divisors remain.

Step-by-Step Solution:
Step 1: Prime factorize 133. 133 = 7 * 19. Step 2: Prime factorize 112. 112 = 2^4 * 7 (that is 16 * 7). Step 3: Identify the common prime factors. Both numbers share the prime factor 7. 133 has factors 7 and 19, while 112 has factors 2^4 and 7. Step 4: Multiply common prime factors using the lowest power. The only common prime is 7, and it appears to the first power in both numbers. Therefore, HCF(133, 112) = 7.

Verification / Alternative check:
To verify, we can simply divide each number by 7. We get 133 / 7 = 19 and 112 / 7 = 16. Both quotients are integers, confirming that 7 is a common divisor. Check if there is any larger common divisor by considering primes above 7; for example, 19 divides 133 but does not divide 112, and higher primes clearly do not divide both. Hence, 7 is indeed the greatest integer that divides both numbers exactly, and therefore it is the HCF.

Why Other Options Are Wrong:
15: This does not divide either 133 or 112 exactly, so it cannot be a common factor at all.
19: While 19 divides 133, it does not divide 112, so it is not a common factor.
6: 6 does not divide 133 exactly, and hence cannot be the HCF.
1: Although 1 divides every integer and is always a common factor, it is not the highest common factor when a larger common factor such as 7 exists.

Common Pitfalls:
Some students may guess based on divisibility by small numbers without systematically checking all possibilities. Others may forget to factorize both numbers properly and miss a shared prime factor. Using clear prime factorization or the Euclidean algorithm keeps the process structured and reduces the chance of mistakes, especially when the numbers become larger or more complex.

Final Answer:
The HCF of 133 and 112 is 7.