Question

Every day a factory making pencils sends $ 2077 $ pencils to the town market and $ 3701 $ pencils to the market in other states. It was found that little more than $ 614 $ pencils remained in the factory. If it is known that the factory makes a perfect number of pencils every day. What is the smallest number of pencils remaining in the factory?

Answer 1

Hint: We are to assume any arbitrary number of pencils that are left in the factory along with the $ 614 $ pencils. Then, we are to suppose the smallest number of pencils that can be made, then we try to find the nearest perfect number to the obtained smallest number of pencils. We are to calculate the smallest number of pencils left in the factory by equating the smallest perfect number we obtained and the smallest number of pencils made that we assumed.

Complete step-by-step answer:
Let us suppose that the smallest number of pencils remaining in the factory along with the $ 614 $ pencils will be $ x $ .
So, the total number of pencils remaining in the factory $ = 614 + x $
Now, given, $ 2077 $ pencils are sent to the town market.
And, $ 3701 $ pencils are sent to the market of other states.
So, the total number of pencils manufactured in the factory $ = (2077 + 3701 + 614 + x) $ pencils
Simplifying the calculations, we get,
 $ = (6392 + x) $ pencils
Now, given, the company makes a perfect number of pencils.
So, the nearest perfect number to $ 6392 $ is $ 6400 $ .
So, it can be clearly said that the factory makes $ 6400 $ pencils every day.
So, we can say,
 $ 6392 + x = 6400 $
Subtracting $ 6392 $ on both sides, we get,
 $ \Rightarrow x = 6400 - 6392 $
 $ \Rightarrow x = 8 $
Therefore, along with the $ 614 $ pencils, more $ 8 $ pencils were left in the factory.
Hence, the smallest number of remaining pencils in factory $ = 614 + 8 = 622 $
So, the correct answer is “8”.

Note: Perfect number is a number that can be expressed as the product of two equal numbers. For example, $ 1,4,9,16 $ etc. In such types of questions, the concept of nearest should be clear, which number to be chosen, whether the question asks for more than a given number or less than that. And if nothing such is mentioned in the question, then the nearest number should be the one that is rounded off to it’s nearest hundred.