What is the largest 4-digit number that is divisible by 20, 50, 60, and 80?

Appeared in: DSSSB - 28 August 2019 (Shift-1)

Explanation

The problem requires finding the largest 4-digit number that is a common multiple of 20, 50, 60, and 80.
First, calculate the Least Common Multiple (LCM) of these numbers. The prime factorization is: 20=2²×5, 50=2×5², 60=2²×3×5, 80=2⁴×5. The LCM is 2⁴×3¹×5² = 16×3×25 = 1200.
The required number must be a multiple of the LCM, 1200.
The largest 4-digit number is 9999. To find the largest multiple of 1200 within this range, divide 9999 by 1200, which gives 8.33.
Taking the integer part (8) and multiplying it by the LCM gives the answer: 1200 × 8 = 9600.

Option B: Although 9000 is a large 4-digit number and divisible by 20, 50, and 60, it is not divisible by 80 (9000 / 80 = 112.5).
Option C: 9240 is not divisible by 50 (a number divisible by 50 must end in 00 or 50) or 80.
Option D: 9480 is not divisible by 50.

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Least Common Multiple (LCM) and Divisibility as background academic context rather than a clinical decision trigger.
This question tests basic arithmetic and logical reasoning, which are essential skills for nurses.
Nurses frequently perform calculations for medication dosages, IV drip rates, and fluid balances, where accuracy is critical for patient safety. Strong numeracy skills are a foundational requirement.
What if the question asked for the smallest 5-digit number divisible by these numbers? In that case, after finding the LCM (1200), you would find the smallest multiple of 1200 that is greater than 9999. 1200 × 9 = 10800.

How to Approach the Question

First, read the question carefully to understand that you need to find a number that is a common multiple of 20, 50, 60, and 80.
Recognize that the most efficient way to solve this is by finding the Least Common Multiple (LCM) of the given divisors.
Calculate the LCM using a method like prime factorization. This gives you the smallest number that all the given numbers can divide into.
Identify the range you are working with, which is the set of 4-digit numbers (1000 to 9999).
Divide the largest number in the range (9999) by the LCM (1200) and take the integer part of the result.
Multiply this integer by the LCM to get the largest multiple within the desired range. This is your answer.

Concept Tested & Keywords

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Practise the full DSSSB - 28 August 2019 (Shift-1)

Attempt every question from this paper in a timed mock, then review the full solution for each one.