Let's look at a simple example:
15:5=3
In this example natural number We divided 15 completely by 3, without remainder.
Sometimes a natural number cannot be completely divided. For example, consider the problem:
There were 16 toys in the closet. There were five children in the group. Each child took the same number of toys. How many toys does each child have?
Solution:
Divide the number 16 by 5 using a column and we get:
We know that 16 cannot be divided by 5. The nearest smaller number that is divisible by 5 is 15 with a remainder of 1. We can write the number 15 as 5⋅3. As a result (16 – dividend, 5 – divisor, 3 – incomplete quotient, 1 – remainder). Received formula division with remainder which can be done checking the solution.
a=
b⋅
c+
d
a – divisible,
b - divider,
c – incomplete quotient,
d - remainder.
Answer: each child will take 3 toys and one toy will remain.
Remainder of division
The remainder must always be less than the divisor.
If during division the remainder is zero, then this means that the dividend is divided completely or without a remainder on the divisor.
If during division the remainder is greater than the divisor, this means that the number found is not the largest. There is a greater number that will divide the dividend and the remainder will be less than the divisor.
Questions on the topic “Division with remainder”:
Can the remainder be greater than the divisor?
Answer: no.
Can the remainder be equal to the divisor?
Answer: no.
How to find the dividend using the incomplete quotient, divisor and remainder?
Answer: We substitute the values of the partial quotient, divisor and remainder into the formula and find the dividend. Formula:
a=b⋅c+d
Example #1:
Perform division with remainder and check: a) 258:7 b) 1873:8
Solution:
a) Divide by column:
258 – dividend,
7 – divider,
36 – incomplete quotient,
6 – remainder. The remainder is less than the divisor 6<7.
7⋅36+6=252+6=258
b) Divide by column:
1873 – divisible,
8 – divisor,
234 – incomplete quotient,
1 – remainder. The remainder is less than divisor 1<8.
Let’s substitute it into the formula and check whether we solved the example correctly:
8⋅234+1=1872+1=1873
Example #2:
What remainders are obtained when dividing natural numbers: a) 3 b)8?
Answer:
a) The remainder is less than the divisor, therefore less than 3. In our case, the remainder can be 0, 1 or 2.
b) The remainder is less than the divisor, therefore less than 8. In our case, the remainder can be 0, 1, 2, 3, 4, 5, 6 or 7.
Example #3:
What is the largest remainder that can be obtained when dividing natural numbers: a) 9 b) 15?
Answer:
a) The remainder is less than the divisor, therefore less than 9. But we need to indicate the largest remainder. That is, the number closest to the divisor. This is the number 8.
b) The remainder is less than the divisor, therefore, less than 15. But we need to indicate the largest remainder. That is, the number closest to the divisor. This number is 14.
Example #4:
Find the dividend: a) a:6=3(rest.4) b) c:24=4(rest.11)
Solution:
a) Solve using the formula:
a=b⋅c+d
(a – dividend, b – divisor, c – partial quotient, d – remainder.)
a:6=3(rest.4)
(a – dividend, 6 – divisor, 3 – partial quotient, 4 – remainder.) Let’s substitute the numbers into the formula:
a=6⋅3+4=22
Answer: a=22
b) Solve using the formula:
a=b⋅c+d
(a – dividend, b – divisor, c – partial quotient, d – remainder.)
s:24=4(rest.11)
(c – dividend, 24 – divisor, 4 – partial quotient, 11 – remainder.) Let’s substitute the numbers into the formula:
с=24⋅4+11=107
Answer: c=107
Task:
Wire 4m. need to be cut into 13cm pieces. How many such pieces will there be?
Solution:
First you need to convert meters to centimeters.
4m.=400cm.
We can divide by a column or in our mind we get:
400:13=30(remaining 10)
Let's check:
13⋅30+10=390+10=400
Answer: You will get 30 pieces and 10 cm of wire will remain.
The article examines the concept of dividing integers with a remainder. Let's prove the theorem on the divisibility of integers with a remainder and look at the connections between dividends and divisors, incomplete quotients and remainders. Let's look at the rules when dividing integers with remainders, looking at them in detail using examples. At the end of the solution we will perform a check.
General understanding of division of integers with remainders
Division of integers with a remainder is considered as a generalized division with a remainder of natural numbers. This is done because natural numbers are a component of integers.
Division with a remainder of an arbitrary number says that the integer a is divided by a number b other than zero. If b = 0, then do not divide with a remainder.
Just like dividing natural numbers with a remainder, integers a and b are divided, with b not zero, by c and d. In this case, a and b are called the dividend and divisor, and d is the remainder of the division, c is an integer or incomplete quotient.
If we assume that the remainder is a non-negative integer, then its value is not greater than the modulus of the number b. Let's write it this way: 0 ≤ d ≤ b. This chain of inequalities is used when comparing 3 or more numbers.
If c is an incomplete quotient, then d is the remainder of dividing the integer a by b, which can be briefly stated: a: b = c (remainder d).
The remainder when dividing numbers a by b can be zero, then they say that a is divisible by b completely, that is, without a remainder. Division without a remainder is considered a special case of division.
If we divide zero by some number, the result is zero. The remainder of the division will also be zero. This can be traced from the theory of dividing zero by an integer.
Now let's look at the meaning of dividing integers with a remainder.
It is known that positive integers are natural numbers, then when dividing with a remainder, the same meaning will be obtained as when dividing natural numbers with a remainder.
Dividing a negative integer a by a positive integer b makes sense. Let's look at an example. Imagine a situation where we have a debt of items in the amount of a that needs to be repaid by b person. To achieve this, everyone needs to contribute equally. To determine the amount of debt for each, you need to pay attention to the value of the private s. The remainder d indicates that the number of items after paying off debts is known.
Let's look at the example of apples. If 2 people owe 7 apples. If we calculate that everyone must return 4 apples, after full calculation they will have 1 apple left. Let us write this as an equality: (− 7) : 2 = − 4 (from t. 1) .
Dividing any number a by an integer does not make sense, but it is possible as an option.
Theorem on the divisibility of integers with remainder
We have identified that a is the dividend, then b is the divisor, c is the partial quotient, and d is the remainder. They are connected to each other. We will show this connection using the equality a = b · c + d. The connection between them is characterized by the divisibility theorem with remainder.
Theorem
Any integer can only be represented through an integer and non-zero number b in this way: a = b · q + r, where q and r are some integers. Here we have 0 ≤ r ≤ b.
Let us prove the possibility of the existence of a = b · q + r.
Proof
If there are two numbers a and b, and a is divisible by b without a remainder, then it follows from the definition that there is a number q, and the equality a = b · q will be true. Then the equality can be considered true: a = b · q + r for r = 0.
Then it is necessary to take q such that given by the inequality b · q< a < b · (q + 1) было верным. Необходимо вычесть b · q из всех частей выражения. Тогда придем к неравенству такого вида: 0 < a − b · q < b .
We have that the value of the expression a − b · q is greater than zero and not greater than the value of the number b, it follows that r = a − b · q. We find that the number a can be represented in the form a = b · q + r.
We now need to consider representing a = b · q + r for negative values of b.
The modulus of the number turns out to be positive, then we get a = b · q 1 + r, where the value q 1 is some integer, r is an integer that meets the condition 0 ≤ r< b . Принимаем q = − q 1 , получим, что a = b · q + r для отрицательных b .
Proof of uniqueness
Let's assume that a = b q + r, q and r are integers with the condition 0 ≤ r true< b , имеется еще одна форма записи в виде a = b · q 1 + r 1 , где q 1 And r 1 are some numbers where q 1 ≠ q, 0 ≤ r 1< b .
When the inequality is subtracted from the left and right sides, then we get 0 = b · (q − q 1) + r − r 1, which is equivalent to r - r 1 = b · q 1 - q. Since the module is used, we obtain the equality r - r 1 = b · q 1 - q.
The given condition says that 0 ≤ r< b и 0 ≤ r 1 < b запишется в виде r - r 1 < b . Имеем, что q And q 1- whole, and q ≠ q 1, then q 1 - q ≥ 1. From here we have that b · q 1 - q ≥ b. The resulting inequalities r - r 1< b и b · q 1 - q ≥ b указывают на то, что такое равенство в виде r - r 1 = b · q 1 - q невозможно в данном случае.
It follows that the number a cannot be represented in any other way except by writing a = b · q + r.
Relationship between dividend, divisor, partial quotient and remainder
Using the equality a = b · c + d, you can find the unknown dividend a when the divisor b with the incomplete quotient c and the remainder d is known.
Example 1
Determine the dividend if, upon division, we get - 21, the partial quotient is 5 and the remainder is 12.
Solution
It is necessary to calculate the dividend a with a known divisor b = − 21, incomplete quotient c = 5 and remainder d = 12. We need to turn to the equality a = b · c + d, from here we get a = (− 21) · 5 + 12. If we follow the order of the actions, we multiply - 21 by 5, after which we get (− 21) · 5 + 12 = − 105 + 12 = − 93.
Answer: - 93 .
The connection between the divisor and the partial quotient and remainder can be expressed using the equalities: b = (a − d) : c , c = (a − d) : b and d = a − b · c . With their help, we can calculate the divisor, partial quotient and remainder. This comes down to constantly finding the remainder when dividing an integer of integers a by b with a known dividend, divisor and partial quotient. The formula d = a − b · c is applied. Let's look at the solution in detail.
Example 2
Find the remainder when dividing the integer - 19 by the integer 3 with a known incomplete quotient equal to - 7.
Solution
To calculate the remainder of division, we apply a formula of the form d = a − b · c. By condition, all data are available: a = − 19, b = 3, c = − 7. From here we get d = a − b · c = − 19 − 3 · (− 7) = − 19 − (− 21) = − 19 + 21 = 2 (difference − 19 − (− 21). This example is calculated using the subtraction rule a negative integer.
Answer: 2 .
All positive integers are natural numbers. It follows that division is performed according to all the rules of division with a remainder of natural numbers. The speed of division with the remainder of natural numbers is important, since not only the division of positive numbers, but also the rules for dividing arbitrary integers are based on it.
The most convenient method of division is a column, since it is easier and faster to get an incomplete or simply a quotient with a remainder. Let's look at the solution in more detail.
Example 3
Divide 14671 by 54.
Solution
This division must be done in a column:
That is, the partial quotient is equal to 271, and the remainder is 37.
Answer: 14,671: 54 = 271. (rest 37)
The rule for dividing with a remainder a positive integer by a negative integer, examples
To perform division with a remainder of a positive number by a negative integer, it is necessary to formulate a rule.
Definition 1
The incomplete quotient of dividing the positive integer a by the negative integer b yields a number that is opposite to the incomplete quotient of dividing the moduli of numbers a by b. Then the remainder is equal to the remainder when a is divided by b.
Hence we have that the incomplete quotient of dividing a positive integer by a negative integer is considered a non-positive integer.
We get the algorithm:
- divide the modulus of the dividend by the modulus of the divisor, then we get an incomplete quotient and
- remainder;
- Let's write down the opposite number to what we got.
Let's look at the example of the algorithm for dividing a positive integer by a negative integer.
Example 4
Divide with remainder 17 by - 5.
Solution
Let's apply the algorithm for dividing with a remainder a positive integer by a negative integer. It is necessary to divide 17 by - 5 modulo. From this we get that the partial quotient is equal to 3, and the remainder is equal to 2.
We get that the required number from dividing 17 by - 5 = - 3 with a remainder equal to 2.
Answer: 17: (− 5) = − 3 (remaining 2).
Example 5
You need to divide 45 by - 15.
Solution
It is necessary to divide the numbers modulo. Divide the number 45 by 15, we get the quotient of 3 without a remainder. This means that the number 45 is divisible by 15 without a remainder. The answer is - 3, since the division was carried out modulo.
45: (- 15) = 45: - 15 = - 45: 15 = - 3
Answer: 45: (− 15) = − 3 .
The formulation of the rule for division with a remainder is as follows.
Definition 2
In order to obtain an incomplete quotient c when dividing a negative integer a by a positive b, you need to apply the opposite of the given number and subtract 1 from it, then the remainder d will be calculated by the formula: d = a − b · c.
Based on the rule, we can conclude that when dividing we get a non-negative integer. To ensure the accuracy of the solution, use the algorithm for dividing a by b with a remainder:
- find the modules of the dividend and divisor;
- divide modulo;
- write the opposite of the given number and subtract 1;
- use the formula for the remainder d = a − b · c.
Let's look at an example of a solution where this algorithm is used.
Example 6
Find the partial quotient and remainder of division - 17 by 5.
Solution
We divide the given numbers modulo. We find that when dividing, the quotient is 3 and the remainder is 2. Since we got 3, the opposite is 3. You need to subtract 1.
− 3 − 1 = − 4 .
The desired value is equal to - 4.
To calculate the remainder, you need a = − 17, b = 5, c = − 4, then d = a − b c = − 17 − 5 (− 4) = − 17 − (− 20) = − 17 + 20 = 3 .
This means that the incomplete quotient of division is the number - 4 with a remainder equal to 3.
Answer:(− 17) : 5 = − 4 (remaining 3).
Example 7
Divide the negative integer - 1404 by the positive 26.
Solution
It is necessary to divide by column and module.
We got the division of the modules of numbers without a remainder. This means that the division is performed without a remainder, and the desired quotient = - 54.
Answer: (− 1 404) : 26 = − 54 .
Division rule with remainder for negative integers, examples
It is necessary to formulate a rule for division with a remainder of negative integers.
Definition 3
To obtain an incomplete quotient c from dividing a negative integer a by a negative integer b, it is necessary to perform modulo calculations, then add 1, then we can perform calculations using the formula d = a − b · c.
It follows that the incomplete quotient of dividing negative integers will be a positive number.
Let us formulate this rule in the form of an algorithm:
- find the modules of the dividend and divisor;
- divide the modulus of the dividend by the modulus of the divisor to obtain an incomplete quotient with
- remainder;
- adding 1 to the incomplete quotient;
- calculation of the remainder based on the formula d = a − b · c.
Let's look at this algorithm using an example.
Example 8
Find the partial quotient and remainder when dividing - 17 by - 5.
Solution
To ensure the correctness of the solution, we apply the algorithm for division with a remainder. First, divide the numbers modulo. From this we get that the incomplete quotient = 3 and the remainder is 2. According to the rule, you need to add the incomplete quotient and 1. We get that 3 + 1 = 4. From here we get that the partial quotient of dividing the given numbers is equal to 4.
To calculate the remainder we will use the formula. By condition we have that a = − 17, b = − 5, c = 4, then, using the formula, we get d = a − b c = − 17 − (− 5) 4 = − 17 − (− 20) = − 17 + 20 = 3 . The required answer, that is, the remainder, is equal to 3, and the partial quotient is equal to 4.
Answer:(− 17) : (− 5) = 4 (remaining 3).
Checking the result of dividing integers with a remainder
After dividing numbers with a remainder, you must perform a check. This check involves 2 stages. First, the remainder d is checked for non-negativity, the condition 0 ≤ d is satisfied< b . При их выполнении разрешено выполнять 2 этап. Если 1 этап не выполнился, значит вычисления произведены с ошибками. Второй этап состоит из того, что равенство a = b · c + d должно быть верным. Иначе в вычисления имеется ошибка.
Let's look at examples.
Example 9
The division is made - 521 by - 12. The quotient is 44, the remainder is 7. Perform check.
Solution
Since the remainder is a positive number, its value is less than the modulus of the divisor. The divisor is - 12, which means its modulus is 12. You can move on to the next check point.
By condition, we have that a = − 521, b = − 12, c = 44, d = 7. From here we calculate b · c + d, where b · c + d = − 12 · 44 + 7 = − 528 + 7 = − 521. It follows that the equality is true. Verification passed.
Example 10
Perform division check (− 17): 5 = − 3 (remaining − 2). Is equality true?
Solution
The point of the first stage is that it is necessary to check the division of integers with a remainder. From this it is clear that the action was performed incorrectly, since a remainder equal to - 2 is given. The remainder is not a negative number.
We have that the second condition is met, but not sufficient for this case.
Answer: No.
Example 11
The number - 19 was divided by - 3. The partial quotient is 7 and the remainder is 1. Check whether this calculation was performed correctly.
Solution
Given a remainder equal to 1. He's positive. The value is less than the divider module, which means that the first stage is being completed. Let's move on to the second stage.
Let's calculate the value of the expression b · c + d. By condition, we have that b = − 3, c = 7, d = 1, which means, substituting the numerical values, we get b · c + d = − 3 · 7 + 1 = − 21 + 1 = − 20. It follows that a = b · c + d the equality does not hold, since the condition gives a = - 19.
From this it follows that the division was made with an error.
Answer: No.
If you notice an error in the text, please highlight it and press Ctrl+Enter
In this article we will look at division of integers with remainder. Let's start with the general principle of dividing integers with a remainder, formulate and prove the theorem on the divisibility of integers with a remainder, and trace the connections between the dividend, divisor, incomplete quotient and remainder. Next, we will outline the rules by which integers are divided with a remainder, and consider the application of these rules when solving examples. After this, we will learn how to check the result of dividing integers with a remainder.
Page navigation.
General understanding of dividing integers with a remainder
We will consider division of integers with a remainder as a generalization of division with a remainder of natural numbers. This is due to the fact that natural numbers are a component of integers.
Let's start with the terms and designations that are used in the description.
By analogy with the division of natural numbers with a remainder, we will assume that the result of division with a remainder of two integers a and b (b is not equal to zero) is two integers c and d. The numbers a and b are called divisible And divider accordingly, the number d – the remainder from dividing a by b, and the integer c is called incomplete private(or just private, if the remainder is zero).
Let us agree to assume that the remainder is a non-negative integer, and its value does not exceed b, that is, (we encountered similar chains of inequalities when we talked about comparing three or more integers).
If the number c is an incomplete quotient, and the number d is the remainder of dividing the integer a by the integer b, then we will briefly write this fact as an equality of the form a:b=c (remaining d).
Note that when dividing an integer a by an integer b, the remainder may be zero. In this case we say that a is divisible by b without a trace(or completely). Thus, division of integers without a remainder is a special case of division of integers with a remainder.
It is also worth saying that when dividing zero by some integer, we are always dealing with division without a remainder, since in this case the quotient will be equal to zero (see the theory section of dividing zero by an integer), and the remainder will also be equal to zero.
We’ve decided on the terminology and notation, now let’s understand the meaning of dividing integers with a remainder.
Dividing a negative integer a by a positive integer b can also be given meaning. To do this, consider a negative integer as debt. Let's imagine this situation. The debt that constitutes the items must be repaid by b people by making an equal contribution. The absolute value of the incomplete quotient c in this case will determine the amount of debt of each of these people, and the remainder d will show how many items will remain after the debt is paid. Let's give an example. Let's say 2 people owe 7 apples. If we assume that each of them owes 4 apples, then after paying the debt they will have 1 apple left. This situation corresponds to equality (−7):2=−4 (remaining 1).
We will not attach any meaning to division with a remainder of an arbitrary integer a by a negative integer, but we will reserve its right to exist.
Theorem on the divisibility of integers with remainder
When we talked about dividing natural numbers with a remainder, we found out that the dividend a, divisor b, partial quotient c and remainder d are related by the equality a=b·c+d. The integers a, b, c and d have the same relationship. This connection is confirmed as follows divisibility theorem with remainder.
Theorem.
Any integer a can be uniquely represented through an integer and non-zero number b in the form a=b·q+r, where q and r are some integers, and .
Proof.
First, we prove the possibility of representing a=b·q+r.
If integers a and b are such that a is divisible by b, then by definition there is an integer q such that a=b·q. In this case, the equality a=b·q+r at r=0 holds.