What Number Divided by 9 Has a Remainder of 8?

When dealing with the question of what number, when divided by 9, gives a remainder of 8, we can utilize modular arithmetic and the division lemma to find a solution. This problem not only involves basic arithmetic but also delves into the deeper realms of number theory and mathematical reasoning. Let's explore the mathematical concepts and derive a general form for such numbers.

Understanding the Problem with Modular Arithmetic

Let's denote the unknown number as ( x ). According to the problem statement, when ( x ) is divided by 9, the remainder is 8. Mathematically, this can be expressed as:

( x equiv 8 mod 9 )

Using the concept of modular arithmetic, we can see that any integer ( x ) that satisfies ( x equiv 8 mod 9 ) is of the form:

( x 9k 8 )

where ( k ) is an integer (positive, negative, or zero).

Examples and Practical Applications

To better understand this, let's calculate a few values of ( x ):

Set ( k 0 ), then ( x 9(0) 8 8 ) Set ( k 1 ), then ( x 9(1) 8 17 ) Set ( k 2 ), then ( x 9(2) 8 26 )

Thus, the numbers 8, 17, 26, and so forth, all satisfy the condition that when divided by 9, they leave a remainder of 8. We can generalize this to say that any number of the form ( 9k 8 ), where ( k ) is a non-negative integer, will meet the given condition.

Using the Division Lemma

The division lemma, or the Euclidean division algorithm, helps us break down the problem further. According to the division lemma:

( text{Dividend} text{Divisor} times text{Quotient} text{Remainder} )

In our context, if we set:

Dividend: ( x ) Divisor: 9 Quotient: ( k ) Remainder: 8

We can derive the following expression:

( x 9k 8 )

Multistep Problem Solving

Let's solve a more complex problem to illustrate the application of these concepts. Given that the divisor is 9, the quotient is 63, and the remainder is 8, we need to find the dividend.

Using the relationship from the division lemma:

( text{Dividend} 9 times 63 8 )

Calculating this, we get:

( text{Dividend} 567 8 575 )

General Formula and Practical Examples

The general formula ( x 9k 8 ) can be applied to find such numbers quickly. For example:

When ( k 1 ): ( x 9(1) 8 17 ) When ( k 2 ): ( x 9(2) 8 26 ) When ( k 3 ): ( x 9(3) 8 35 )

Another practical example is finding the smallest 3-digit multiple of 9 minus 1. To find this, we:

Identify the smallest 3-digit multiple of 9, which is 108. Subtract 1 from 108: ( 108 - 1 107 ).

Thus, 107 is the smallest 3-digit number that, when divided by 9, leaves a remainder of 8.

By understanding modular arithmetic and the division lemma, we can easily solve problems involving remainders and divisors, making complex calculations straightforward and accessible.