Question

Atleast what number must be subtracted from 434079 so that it becomes divisible by \(137 ?\) (a) 173 (b) 63 (c) 97 (d) can't be determined.

Short answer

Based on the step-by-step solution above, the short answer is: To make the number 434079 divisible by 137, you must subtract 97 from it.

Step-by-step solution

Find the remainder when 434079 is divided by 137

To find the remainder, we can use the formula: remainder = dividend - (quotient x divisor). First, divide 434079 by 137 and find the quotient (ignoring any remaining decimals). Quotient = 434079 ÷ 137 = 3167 (ignoring the decimal points)

Calculate the remainder using the formula

Now use the formula above to find the remainder: remainder = 434079 - (3167 x 137) remainder = 434079 - 433982 remainder = 97

Determine the number to be subtracted

Since the remainder when 434079 is divided by 137 is 97, this means we need to subtract 97 from 434079 in order to make it divisible by 137. So, the correct option is (c) 97.

Key concepts

Remainder Calculation

When working with whole numbers, finding the remainder of a division problem is a key concept in arithmetic. It's essentially the amount left over after dividing a number into equal parts. For example, if you divide 10 by 3, you can fit the number 3 into 10 three times, which equals to 9, with 1 left over as the remainder.

To calculate the remainder manually, you can use the subtraction method, where you multiply the divisor by a whole number that gets you as close as possible to the dividend without going over, and then subtract this from the dividend. This gives you the remainder. For instance, with our earlier example, the math goes as follows: remainder = 10 - (3 x 3) = 10 - 9 = 1.

Using this method requires a good understanding of multiplication and subtraction. You must be able to quickly multiply the divisor by different numbers to find the quotient that brings you closest to the dividend. In our original exercise, after finding out that the quotient of 434079 divided by 137 is 3167, we apply the formula: remainder = 434079 - (3167 x 137) to find that the remainder is 97.

Arithmetic Division

Division is one of the basic operations in arithmetic. It splits a number (the dividend) into equal parts (as described by the divisor) and identifies how many times the divisor fits into the dividend. The result is called the quotient. If the divisor does not perfectly fit into the dividend, the leftover part is referred to as the remainder.

Performing Division

Arithmetic division can be done in several ways, including long division, short division, or by using a calculator. Long division is the standard method taught in school for dividing larger numbers. In the context of the exercise, we can simplify the division for ease of understanding by ignoring any fractional parts of the quotient. Here the quotient of 434079 divided by 137 is 3167, a whole number, leaving out any fractional component that would be part of a more precise calculation.

Understanding that arithmetic division is not always 'clean' (meaning it doesn't always result in a whole number as the quotient) sets the stage for realizing the practicality of remainder calculations in scenarios such as equal distribution of items or checks for divisibility.

Number Theory

Number theory is the branch of mathematics that deals with the properties and relationships of numbers, particularly integers. It delves into various topics including primes, divisibility, and, importantly for our exercise, remainders. The theory underpins the concepts and techniques used to solve divisibility problems.

What Number Theory Involves

Many number theory problems involve divisibility and remainders, just like the textbook exercise in question. It's essential when working with number theory to be familiar with various divisibility rules, which are shortcuts that help determine if one number is divisible by another without performing the entire division process.

Applying number theory to our exercise is about understanding that for any integer to be divisible by another, the remainder must be zero. The understanding that 434079 needs to be subtracted by 97 to be divisible by 137 comes from the number theory's fundamental rule about remainders: a number is divisible by another if, and only if, the remainder after dividing these numbers is zero.