In aptitude (HCF / GCD simplification), find the greatest number that will divide 390, 495, and 300 exactly without leaving any remainder, that is, determine the highest common factor of the three given integers.

Introduction / Context:
This question checks basic number theory skills related to the highest common factor (HCF), also called greatest common divisor (GCD). Many aptitude tests include similar problems where candidates must find the largest integer that divides multiple numbers exactly. Mastery of prime factorisation or repeated division is essential for quick and accurate answers.

Given Data / Assumptions:
- The three numbers are 390, 495, and 300.
- We want the greatest integer that divides all three without leaving any remainder.
- This integer is the HCF or GCD of the three numbers.

Concept / Approach:
The HCF of several numbers is the largest number that divides each of them exactly. There are two popular methods: prime factorisation and successive division (Euclidean algorithm). For these moderate-sized numbers, prime factorisation is straightforward. We factor each number into primes and then multiply the common prime factors with the smallest exponents.

Step-by-Step Solution:
Step 1: Prime factorise 390.390 = 39 * 10 = 3 * 13 * 2 * 5 = 2 * 3 * 5 * 13.Step 2: Prime factorise 495.495 = 49.5 * 10, but better: 495 = 5 * 99 = 5 * 9 * 11 = 5 * 3^2 * 11.Step 3: Prime factorise 300.300 = 3 * 100 = 3 * 4 * 25 = 3 * 2^2 * 5^2 = 2^2 * 3 * 5^2.Step 4: Identify common prime factors across all three numbers.Common primes: 3 and 5 appear in all three factorizations.Step 5: For prime 3, the smallest exponent among the numbers is 1 (since 390 has 3^1, 495 has 3^2, and 300 has 3^1).Step 6: For prime 5, the smallest exponent among the numbers is 1 (since 390 has 5^1, 495 has 5^1, and 300 has 5^2).Step 7: So HCF = 3^1 * 5^1 = 3 * 5 = 15.

Verification / Alternative check:
Verify by direct division: 390 / 15 = 26, 495 / 15 = 33, and 300 / 15 = 20. All quotients are integers, and 15 divides each number exactly. Check that no larger option divides all three: for example, 25 does not divide 390 exactly (390 / 25 is not an integer), and 30 does not divide 495 exactly (495 / 30 is not an integer). Therefore, 15 is indeed the greatest common divisor.

Why Other Options Are Wrong:
- 5: This is a common divisor but not the greatest one, since 15 is larger and still divides all three numbers exactly.
- 25: It divides 300 and 495 is not divisible by 25, so it fails the condition.
- 35: This does not divide 300 or 390 exactly; it is not common.

- 30: It divides 390 and 300, but 495 / 30 is not an integer, so it is not a common divisor of all three numbers.

Common Pitfalls:
Students may quickly pick a small common factor like 5 or 3 without checking whether it is the greatest. Others might incorrectly factor the numbers or miss a higher common factor such as 15. It is important to systematically list the prime factors and then combine only those that appear in all numbers with the smallest exponent. A quick divisibility check for each option can also help confirm the correct answer in multiple choice settings.

Final Answer:
The greatest number that divides 390, 495, and 300 exactly is 15.