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The division of integers
During a board game, Henry stepped on the lucky field, so he could divide his \latex{ 15 } loan notes into three equal parts. How many loan notes did each child receive? How did their financial situation change in the game?
Example 1
Perform the following divisions.
- \latex{(-15)\div(+3)}
- \latex{(-15)\div(-3)}
Solution
- \latex{(-15)\div(+3) = \Diamond.}
Find the number which gives \latex{ (–15) } when multiplied by \latex{ (+3) }, that is,
\latex{(+3)\times\Diamond=-15.}
This number is \latex{ (–5) } because \latex{ (+3) × (–5) = –15 }.
Therefore
\latex{(-15)\div(+3) = -5.}
- \latex{(-15)\div(-3) =\triangle.}
Find the number which gives \latex{ (–15) } when multiplied by \latex{ (–3) }, that is,
\latex{(-3)\times\triangle = -15.}
This number is \latex{ (+5) } because \latex{ (–3) × (+5) = –15. }
Therefore
\latex{(-15)\div(-3) = +5.}
In Example 1, the division of integers was performed by using multiplication.
Observe how the sign changes depending on the sign of the dividend and the divisor.
\latex{+4}
\latex{+4}
\latex{-4}
\latex{-4}
\latex{+20}
\latex{+20}
\latex{-20}
\latex{-20}
\latex{\times (+5)}
\latex{\times (+5)}
\latex{\times (-5)}
\latex{\times (-5)}
\latex{\div (-5)}
\latex{\div (-5)}
\latex{\div (+5)}
\latex{\div (+5)}
\latex{(+4)\times(+5) = (+20),}
\latex{(\textcolor{red}{+}20)\div(\textcolor{red}{+}5) = (\textcolor{red}{+}4).}
\latex{(+4)\times(-5) = (-20),}
\latex{(\textcolor{009fe3}{-}20)\div(\textcolor{009fe3}{-}5) = (\textcolor{red}{+}4).}
\latex{(-4)\times(+5) = (-20),}
\latex{(\textcolor{009fe3}{-}20)\div(\textcolor{red}{+}5) = (\textcolor{009fe3}{-}4).}
\latex{(-4)\times(-5) = (+20),}
\latex{(\textcolor{red}{+}20)\div(\textcolor{009fe3}{-}5) = (\textcolor{009fe3}{-}4).}
thus
thus
thus
thus
The quotient of two numbers with the same sign is a positive number, while the quotient of two numbers with different signs is a negative number.
The absolute value of a quotient can be calculated by dividing the absolute value of the dividend by that of the divisor.
Zero in the division of integers
When \latex{ 0 } is divided by an integer (other than zero), the quotient is zero.
\latex{ 0 ÷ (–8) = 0 } because \latex{ 0 × (–8) = 0 }.
The divisor cannot be \latex{0}.
- \latex{ (–9) ÷ 0 = △ } would mean that \latex{ △ × 0 = –9 }, but there is no such number, as multiplying any number by zero is zero.
- \latex{ 0 ÷ 0 = □ } would mean that \latex{□×0=0}, but multiplying any number by \latex{ 0 } is \latex{ 0 }, so any number can be written instead of □. The quotient cannot be determined accurately; therefore, \latex{ 0 ÷ 0 } is undefined.
Dividing integers by \latex{ 0 } cannot be defined.
Exercises
{{exercise_number}}. Compare the quotients.
- \latex{(+72)\div(+6)\quad (+72)\div(+3)}
- \latex{(+72)\div(-6)\quad (+72)\div (-3)}
- \latex{(-72)\div(+6)\quad (+72)\div (+3)}
- \latex{(-72)\div(-6)\quad(+72)\div(-3)}
- \latex{(-72)\div(-6)\quad(-72)\div(+3)}
- \latex{(-72)\div(+6)\quad(-72)\div(+3)}
{{exercise_number}}. Arrange the quotients in ascending order.
A
B
C
D
E
F
\latex{(-84)\div (-4)}
\latex{(+64)\div(-2)}
\latex{(-32)\div(-4)}
\latex{(+64)\div(-8)}
\latex{(-64)\div(+4)}
\latex{(-32)\div(+2)}
{{exercise_number}}. Arrange the quotients in descending order.
- \latex{\fcolorbox{f6b900}{fef6e3}{$(-108)\div(+36)$}}
- \latex{\fcolorbox{f6b900}{fef6e3}{$(+72)\div(-36)$}}
- \latex{\fcolorbox{f6b900}{fef6e3}{$(-108)\div(-36)$}}
- \latex{\fcolorbox{f6b900}{fef6e3}{$(-108)\div(-18)$}}
- \latex{\fcolorbox{f6b900}{fef6e3}{$(+108)\div(+54)$}}
- \latex{\fcolorbox{f6b900}{fef6e3}{$(-54)\div(-18)$}}
{{exercise_number}}. Calculate the missing values if \latex{\large{ a × b} = –256 }.
\latex{a}
\latex{b}
\latex{-8}
\latex{+16}
\latex{-1}
\latex{+64}
\latex{-2}
\latex{+2}
\latex{-32}
\latex{-16}
\latex{+4}
{{exercise_number}}. Fill in the missing numbers.
\latex{(-144)\div\square = (+72)\div\square = (-36)\div(+3) = -12\div\square}
{{exercise_number}}. Determine which statement is true based on the quotient of \latex{ (–54) } and \latex{ (+18) }.
- If the dividend is multiplied by \latex{ –1 }, the quotient becomes \latex{ –1 } times greater.
- If the dividend and the divisor are divided by \latex{ +2 }, the quotient remains unchanged.
- If \latex{ –6 } is added to the dividend and the divisor, the quotient remains unchanged.
{{exercise_number}}. What numbers can be divided by
- \latex{ –60 } to get integers greater than \latex{ +1 };
- \latex{ +36 } to get integers smaller than \latex{ -2 };
- \latex{+75} to get integers divisible by \latex{5?}
{{exercise_number}}. How much should be subtracted from the quotient of
- \latex{-51} and \latex{+17} to get \latex{-10;}
- \latex{+57} and \latex{-3} to get \latex{+10;}
- \latex{-81} and \latex{-9} to get \latex{-9?}
{{exercise_number}}.
- How many times \latex{ 18 } is equal to the product of \latex{ –12 } and \latex{ –24 }?
- \latex{ –2,730 } is how many times the product of \latex{ –13 } and \latex{ –14 }?
- How many times \latex{ 11 } is equal to the quotient of \latex{ 1,716 } and \latex{ –13 }?
{{exercise_number}}. By how much is the quotient of
- \latex{ –60 } and \latex{ +10 } multiplied by if you get an integer smaller than \latex{ 10 };
- \latex{ +50 } and \latex{ –10 } multiplied by if you get a two-digit positive integer;
- \latex{ +180 } and \latex{ –45 } multiplied by if you get \latex{ 0 ?}
{{exercise_number}}. Which integer is it?
- one-fourth of \latex{-24}
- one-fourth of which is \latex{ –24 }
- \latex{ –4 } times the integer is \latex{ –24 }
- the integer divided by \latex{ –4 } is \latex{ –24 }
{{exercise_number}}. Choose a single-digit positive number. Write it down nine times. Divide the resulting nine-digit number by \latex{ –9 }. Divide the quotient by the additive inverse of the chosen number. What do you notice? Repeat the exercise with another freely chosen single-digit number.
{{exercise_number}}. Add the sum of its digits to \latex{ –999 }, then divide the result by \latex{ –9 }. Is the resulting number divisible by \latex{ 9 ?}
{{exercise_number}}. Perform the divisions.
- \latex{(+72)\div[(+6)\div(+3)]}
- \latex{[(+16)\div(+8)]\div[(-2)\div 0]}
- \latex{[(-54)\div(-9)]\div[(-3) \div(-1)]}
- \latex{[(-49)\div(-7)]\div(+7)}
- \latex{[(-54)\div(-3)]\div[(+6) \div(-3)]}
- \latex{0\div(+72)\div(-9)}
{{exercise_number}}.
- Divide \latex{2,520} by \latex{-7}, the result by \latex{-8}, then the resulting quotient by \latex{-9}. Divide \latex{2,520} by
the product of \latex{-7}, \latex{-8} and \latex{-9}. Compare the two results.
{{exercise_number}}. The results of which divisions are equal?
- \latex{(-39)\div(-13)}
- \latex{(+68)\div(-20 + 3)}
- \latex{(-3 -57)\div(-3 + 18)}
- \latex{(-170)\div(+10)}
- \latex{(-63-37)\div(+2)}
- \latex{[-20\times(-3-2)]\div(-2)}
- \latex{(-51)\div(+3)}
- \latex{(+135)\div(+45)}
- \latex{[-27 + (-9)]\div(-12)}
{{exercise_number}}. Which one is greater? Pay attention to the order of operations.
- \latex{(-100)\div(-5) + (-5) \quad \text{or} \quad(-100)\div[-5 + (-5)]}
- \latex{-24-80\div(-8) \quad \text{or}\quad (-24-80)\div(-8)}
- \latex{+36 + (-60)\div(-12)\quad \text{or} \quad[+36 + (-60)]\div(-12)}
{{exercise_number}}. Draw a coordinate system. Draw a triangle with vertices \latex{ A(+2; +12) }, \latex{ B(–8; –4) } and \latex{ C(+4; 0) }. Use the same coordinate system but different colours to draw triangles whose vertices can be determined in the following way:
- the first coordinates are multiplied by \latex{ –1 }, while the second coordinates remain unchanged;
- the first coordinates remain unchanged, while the second coordinates are divided by \latex{ –2 };
- both coordinates are divided by two.
Quiz
What is the smallest natural number divisible by every integer from \latex{ 1 } to \latex{ 10 ?}