Question

In the above question, at least what number be added to 434079 , so that it will become divisible by (or multiple of) \(137 ?\) (a) 97 (b) 74 (c) 75 (d) none of these

Short answer

Answer: None of these (The smallest number is 85)

Step-by-step solution

Divide 434079 by 137

Divide 434079 by 137 to find the quotient and the remainder: $$ 434079 \div 137 = 3167 \text{ (quotient) with a remainder of } 52. $$

Subtract the remainder from 137

To find the smallest number that needs to be added to 434079 to make it divisible by 137, subtract the remainder obtained in step 1 from 137: $$ 137 - 52 = 85. $$

Match the answer with the given options

Now we can see that 85 is the smallest number that can be added to 434079 to make it divisible by 137; however, this number is not available in the given options. Therefore, the correct answer is: (d) none of these

Key concepts

Remainder

When dealing with division, the remainder is what is "left over" after a number has been divided as completely as possible by another number. For example, when dividing 434,079 by 137, we reach a point where 3167 complete groups of 137 fit within 434,079, leaving behind a remainder of 52. This remainder, 52, represents the part of 434,079 that cannot be evenly divided by 137.

The remainder is a crucial part of the division process that helps determine how close a number is to being a perfect multiple of the divisor. In problems where you need to make a number divisible by another, knowing the remainder can tell you exactly how much more is needed to reach the nearest multiple.

• Remainder is always less than the divisor.
• In mathematical notation, if a number "a" is divided by "b", then the expression can be written as: \(a = bq + r\), where "q" is the quotient and "r" is the remainder.
• Remainder can be useful in modular arithmetic, number games, and solving problems related to divisibility.

Quotient

The quotient is the result you get when you divide one number by another. In the example of dividing 434,079 by 137, the quotient is 3,167. This means that we can fit 3,167 complete groups of 137 into the number 434,079.

Understanding the quotient is essential because it shows how many times the divisor fits completely into the dividend without exceeding it. Quotients are used in many areas such as simplifying problems, algebra, and real-world applications like finding average values.

• The quotient is the number of times the divisor is contained within the dividend.
• It's an integral part of the equation \(a = bq + r\), guiding us in understanding the relationship between all parts of a division problem.
• An accurate quotient helps solve larger problems that involve managing resources or dividing tasks.

Multiple

A multiple of a number is the product of that number and an integer. In the context of the problem, we need 434,079 to become a multiple of 137 by adding just the right number to it.

To find this multiple, we first considered the closest product of 137 below 434,079 and then found the difference needed to reach the next multiple, which in this case was determined after noticing the remainder from the division. Multiples are fundamental in arithmetic as they help in understanding divisibility and in solving problems where numbers are grouped or arranged in specific ways.

• A number is a multiple of another if it can be expressed in the form \(n imes m\), where both "n" and "m" are integers.
• Multiples have fixed intervals; every new multiple of a number is just an increment by that number over the previous multiple.
• Finding multiples is essential for tasks like scheduling, repetitive events, and understanding mathematical patterns.