Solving for the Smallest Number for Divisibility by 58

The problem asks us to find the smallest positive integer that needs to be added to the number 57846 so that the resulting sum is exactly divisible by 58. To solve this, we need to understand the concept of divisibility and remainders.

Understanding Divisibility and Remainders

A number is completely divisible by another number if, upon division, the remainder is 0. If a number, say '$N$', is divided by a divisor '$d$', we can express it as: $N = q \times d + r$ where '$q$' is the quotient and '$r$' is the remainder, with $0 \le r < d$.

In this case, our number $N$ is 57846 and the divisor $d$ is 58.

Step-by-Step Calculation

1. Divide the given number by the divisor: We need to divide 57846 by 58.

   $57846 \div 58$

2. Calculate the quotient and remainder: Performing the division:

   $57846 = 997 \times 58 + 20$

   Here, the quotient ($q$) is 997 and the remainder ($r$) is 20.

3. Determine the number to add: Since the remainder is 20, 57846 is 20 more than a multiple of 58 (which is $997 \times 58 = 57826$). To get the next multiple of 58, we need to add the difference between the divisor (58) and the remainder (20) to the original number.

   The smallest number to add is calculated as:

   $ \text{Number to add} = \text{Divisor} - \text{Remainder} $ $ \text{Number to add} = 58 - 20 $ $ \text{Number to add} = 38 $

4. Verify the result: If we add 38 to 57846, we get:

   $ 57846 + 38 = 57884 $

   Now, let's check if 57884 is divisible by 58:

   $ 57884 \div 58 = 998 $

   Since the division results in a whole number (998) with no remainder, 57884 is completely divisible by 58. Therefore, the smallest number that should be added is 38.

Conclusion

The smallest number that must be added to 57846 to make it perfectly divisible by 58 is 38.