What is the least number that should be added to 2560 so that the sum is exactly divisible by 3, 5, 8 and 15?

DSSSB - 29 August 2019 (Shift-1)

Non Nursing Subjects

Hard

Appeared in: DSSSB - 29 August 2019 (Shift-1)

Explanation

The core concept is that for a number to be divisible by a set of other numbers, it must be a multiple of their Least Common Multiple (LCM).
First, calculate the LCM of the divisors (3, 5, 8, 15). The prime factorization is 2³, 3¹, and 5¹. Thus, the LCM is 2³ × 3 × 5 = 120.
Next, divide the given number (2560) by the LCM (120). The division 2560 ÷ 120 gives a quotient of 21 and a remainder of 40.
The number that needs to be added is the difference between the LCM and the remainder. This is calculated as 120 - 40 = 80.
Adding 80 to 2560 gives 2640. This sum (2640) is perfectly divisible by 120 (2640 / 120 = 22), and therefore also by 3, 5, 8, and 15.

Option B: Adding 100 to 2560 results in 2660. When 2660 is divided by the LCM (120), the remainder is 20 (2660 = 120 × 22 + 20). Since the remainder is not zero, it is not exactly divisible.
Option C: Adding 60 to 2560 results in 2620. When 2620 is divided by the LCM (120), the remainder is 100 (2620 = 120 × 21 + 100). Since the remainder is not zero, it is not exactly divisible.
Option D: Adding 40 to 2560 results in 2600. When 2600 is divided by the LCM (120), the remainder is 80 (2600 = 120 × 21 + 80). Since the remainder is not zero, it is not exactly divisible. Note that 40 is the remainder from the initial division, not the number to be added.

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Divisibility and Least Common Multiple (LCM) as background academic context rather than a clinical decision trigger.
This question tests general aptitude and numerical ability, which is a component of many nursing entrance examinations.
Strong numerical skills are essential for nurses for tasks like medication dosage calculation, IV drip rate monitoring, and interpreting patient data, ensuring patient safety and effective care.
What if? - If the stem changed one defining clue so that 80 no longer matched, reassess the option whose mechanism, classification, or indication now fits the revised presentation.

How to Approach the Question

Identify the goal: Find the smallest number to add to 2560 to make it divisible by 3, 5, 8, and 15.
Recognize the key concept: A number divisible by several others must be a multiple of their Least Common Multiple (LCM).
Step 1: Calculate the LCM of the divisors (3, 5, 8, 15), which is 120.
Step 2: Divide the given number (2560) by the LCM (120) and find the remainder. (2560 ÷ 120 gives a remainder of 40).
Step 3: To find the number to add, subtract the remainder from the LCM (120 - 40 = 80).
Verify the result: 2560 + 80 = 2640. Check if 2640 is divisible by 120. It is (2640 / 120 = 22). Select the corresponding option.

Concept Tested & Keywords

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