What is the least four-digit natural number that is divisible by 4, 6, 8 and 10?

Introduction / Context:
This problem is about finding the smallest four-digit number that is exactly divisible by several given numbers. This is a typical application of the least common multiple (LCM), which is a core concept in arithmetic and number systems. Being comfortable with LCM helps solve many divisibility and scheduling type questions that appear frequently in aptitude exams.

Given Data / Assumptions:

• We are looking for a four-digit number (that is, between 1000 and 9999).
• The number must be divisible by 4, 6, 8 and 10.
• We want the least (smallest) such four-digit number.

Concept / Approach:
A number divisible by several integers must be divisible by their least common multiple. Therefore, we first find the LCM of 4, 6, 8 and 10. Once we have the LCM, any common multiple of all four numbers is a multiple of this LCM. We then look for the smallest multiple of the LCM that is at least 1000. This process ensures that the number we find is the smallest four-digit number that satisfies all the divisibility conditions.

Step-by-Step Solution:
Step 1: Factorise each number into primes. 4 = 2^2, 6 = 2 * 3, 8 = 2^3, 10 = 2 * 5. Step 2: For the LCM, take the highest power of each prime that appears in any factorisation. For prime 2, the highest power is 2^3 from 8. For prime 3, the highest power is 3 from 6. For prime 5, the highest power is 5 from 10. Step 3: Multiply these highest powers: LCM = 2^3 * 3 * 5 = 8 * 3 * 5 = 24 * 5 = 120. Step 4: Now find the smallest multiple of 120 that is at least 1000. Compute 1000 / 120 which is about 8.33. Step 5: The next whole number after 8.33 is 9, so the required multiple is 9 * 120 = 1080.

Verification / Alternative check:
Check divisibility of 1080 by each given number. 1080 / 4 = 270 (integer), 1080 / 6 = 180 (integer), 1080 / 8 = 135 (integer), and 1080 / 10 = 108 (integer). So 1080 is divisible by all four numbers. Also, the previous multiple of 120 is 8 * 120 = 960, which is only a three-digit number. Therefore, 1080 is indeed the smallest four-digit number that satisfies the condition.

Why Other Options Are Wrong:
Option 1107: This is not divisible by 4 or 8 and so fails the conditions.
Option 1100: While divisible by 4 and 10, it is not divisible by 8 or 6.
Option 1208 and 1040: These may be divisible by some of the given numbers, but they are not all divisible by 4, 6, 8 and 10 together or they are not the smallest that satisfies all conditions.

Common Pitfalls:
One common mistake is to check the options one by one without first calculating the LCM, which can be time consuming and error prone. Another error is to forget to check all the divisibility conditions or to ignore the “least” and “four-digit” parts of the question. Always start such problems by finding the LCM, and then jump directly to the smallest required multiple above the lower limit of the range.

Final Answer:
The least four-digit number that is divisible by 4, 6, 8 and 10 is 1080.