Question

At least which number must be subtracted from 9999999 so that it will become the multiple of \(125 ?\) (a) 124 (b) 4 (c) 24 (d) none of these

Short answer

Answer: Although 26 is not an option, it is the correct answer.

Step-by-step solution

Find the remainder when 9999999 is divided by 125

Divide 9999999 by 125 and find the remainder: Remainder = 9999999 mod 125

Calculate the difference between 125 and the remainder

Subtract the remainder obtained in step 1 from 125 since we want a number which we subtract from 9999999 to get its multiple. Difference = 125 - Remainder

Compare the difference with the options provided

Check if the calculated difference is in the given options. If it is, then that option is the correct answer. Let's perform the calculations.

Calculation

Remainder = 9999999 mod 125 = 99

Calculation

Difference = 125 - 99 = 26 Since the difference is 26, we can see that it is not one of the options but 26 can be represented as (24 + 2), so option (c) 24 can be part of the correct answer, therefore the motivation behind adding the remaining 2 to our answer is somewhat forced. Using this information, let's check if subtracting 24 from 9999999 results in a multiple of 125.

Key concepts

Modular Arithmetic

Modular Arithmetic is like clock arithmetic. It deals with numbers wrapping around after a certain value, called the modulus. To understand it, think about a clock: after 12 comes back 1 again. In mathematics, the modulus is the number at which we wrap around. For example, in a 12-hour clock, 12 is the modulus.

When we see a statement such as \( 9999999 \mod 125 \), it means if you divide 9999999 by 125, what would the remainder be? This helps to determine how far a number is from being a multiple of the modulus (in this case, 125).

• For the number 9999999, when divided by 125, the remainder is 99.
• This tells us 9999999 is 99 higher than the nearest lower multiple of 125.

This remainder is crucial to solving many arithmetic problems, like finding out which numbers to subtract to reach a clean multiple of 125.

Divisibility

In mathematics, the concept of divisibility is straightforward yet powerful. A number is divisible by another if, when divided, the remainder is zero. It's the idea that one number can become a perfect part of another without leftovers, much like how slices of a pie can divide a pie evenly.

For example, a number is divisible by 125 if it can be written as \( 125k \) (where \( k \) is an integer). To check if a large number such as 9999999 is divisible by 125, we look for its remainder when divided by 125.

• In this scenario, since 9999999 modulo 125 gives a remainder of 99, it's clear that 9999999 isn't directly divisible by 125.

Understanding divisibility helps in finding the closest number which is divisible, simplifying many arithmetic tasks.

Remainder Theorem

The Remainder Theorem is a useful rule in number theory, which gives insights into the remainder when a polynomial is divided by a certain number. For integers, instead of polynomials, it aids in understanding how far off a number is from another divisor.

Continuing with our example, we calculate \( 9999999 \mod 125 \) and identify 99 as the remainder. The goal is to make this remainder zero, which can be achieved by subtracting the remainder from the number.

• Subtract the remainder (99 in this case) from the modulus (125).
• This gives \( 125 - 99 = 26 \). The next multiple of 125 can be achieved by subtracting 26 from 9999999.

Sometimes, as seen here, solutions may require excluding certain options or identifying alternative values that provide a solution. This adaptive approach stems from the remainder theorem, encouraging flexible thinking in mathematical problem-solving.