DIVISIONS IN WHICH THE REMAINDER IS 300: HOW THEY ARE BUILT - MATHS - 2025

There are many divisions in which the remainder is 300. In addition to citing some of them, a technique will be shown that helps to build each of these divisions, which does not depend on the number 300.

This technique is provided by the Euclidean division algorithm, which states the following: given two integers "n" and "b", with "b" different from zero (b ≠ 0), there are only integers "q" and «R», such that n = bq + r, where 0 ≤ «r» <-b-.

Euclid's division algorithm

The numbers "n," "b," "q," and "r" are called dividend, divisor, quotient, and remainder (or remainder), respectively.

It should be noted that by requiring that the remainder be 300, it is implicitly saying that the absolute value of the divisor must be greater than 300, that is: -b-> 300.

Some divisions in which the remainder is 300

Here are some divisions in which the remainder is 300; then, the construction method of each division is presented.

1- 1000 ÷ 350

If you divide 1000 by 350, you can see that the quotient is 2 and the remainder is 300.

2- 1500 ÷ 400

Dividing 1500 by 400, the quotient is 3 and the remainder is 300.

3- 3800 ÷ 700

By doing this division, the quotient will be 5 and the remainder will be 300.

4- 1350 ÷ (−350)

When this division is solved, we obtain -3 as a quotient and 300 as a remainder.

How are these divisions built?

To build the previous divisions it is only necessary to use the division algorithm properly.

The four steps to build these divisions are:

1- Fix the Residue

Since we want the remainder to be 300, we set r = 300.

2- Choose a divisor

Since the remainder is 300, the divisor to be chosen must be any number such that its absolute value is greater than 300.

3- Choose a quotient

For the quotient, you can choose any integer other than zero (q ≠ 0).

4- The dividend is calculated

Once the remainder, divisor, and quotient are set, they are substituted on the right side of the division algorithm. The result will be the number to be chosen as the dividend.

With these four easy steps you can see how each division in the list above was built. In all these, r = 300 was set.

For the first division, b = 350 and q = 2 were chosen. Substituting in the division algorithm gave the result 1000. So the dividend must be 1000.

For the second division, b = 400 and q = 3 were established, so that when substituting in the division algorithm, 1500 was obtained. Thus, it is established that the dividend is 1500.

For the third, the number 700 was chosen as the divisor and the number 5 as the quotient. When evaluating these values ​​in the division algorithm, it was obtained that the dividend must be equal to 3800.

For the fourth division, the divisor equal to -350 and the quotient equal to -3 were set. When these values ​​are substituted in the division algorithm and solved, it is obtained that the dividend is equal to 1350.

By following these steps you can build many more divisions where the remainder is 300, being careful when using negative numbers.

It should be noted that the construction process described above can be applied to construct divisions with residuals other than 300. Only the number 300, in the first and second steps, is changed to the desired number.

References

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2. Eisenbud, D. (2013). Commutative Algebra: with a View Toward Algebraic Geometry (llustrated ed.). Springer Science & Business Media.
3. Johnston, W., & McAllister, A. (2009). A Transition to Advanced Mathematics: A Survey Course. Oxford University Press.
4. Penner, RC (1999). Discrete Mathematics: Proof Techniques and Mathematical Structures (illustrated, reprint ed.). World Scientific.
5. Sigler, LE (1981). Algebra. Reverte.
6. Zaragoza, AC (2009). Number Theory. Vision Books.