What is the highest common factor (HCF) of the integers 133 and 112?

Introduction:
This question is about finding the highest common factor (HCF) or greatest common divisor (GCD) of two numbers, 133 and 112. It helps reinforce mental factorization skills that are important in many quantitative aptitude topics such as fractions, ratios and simplification.

Given Data / Assumptions:

• First number = 133.
• Second number = 112.
• We need to determine their HCF.

Concept / Approach:
The HCF of two numbers is the largest positive integer that divides both of them exactly. We can find it either by prime factorization or by the Euclidean algorithm. Here, prime factorization is quite simple.

Step-by-Step Solution:
Step 1: Factorize 133.133 = 7 * 19Step 2: Factorize 112.112 = 16 * 7 = 2^4 * 7Step 3: Identify common prime factors.Common prime factor is 7.Step 4: Compute the HCF.Since 7 is the only common prime factor and appears to the power 1 in both, HCF = 7

Verification / Alternative check:
We can verify by direct division. 133 ÷ 7 = 19, an integer, and 112 ÷ 7 = 16, also an integer. Try any larger candidate such as 14 or 19: 133 is not divisible by 14, and 112 is not divisible by 19. Hence no number larger than 7 divides both numbers, confirming that 7 is indeed the highest common factor.

Why Other Options Are Wrong:
15 and 6: Neither of these divides 133 exactly, so they cannot be the HCF.
19: While 19 divides 133, it does not divide 112, so it cannot be a common factor.
14: 14 divides 112 but not 133, so it is also not a common factor of both.

Common Pitfalls:
Some students check small divisors randomly and may miss the correct one if they do not factorize systematically. Others think a larger factor like 19 or 14 may work without verifying divisibility for both numbers. Always factor or apply the Euclidean algorithm to be sure.

Final Answer:
The highest common factor of 133 and 112 is 7.