Ostukorv on tühi
Look at the remainder
11 kayakers go rowing on a river. They have a choice to rent, 1, 2, 3 or 4-person kayaks. Can they rent only one type of kayak so that there are the same number of people in each kayak?
\latex{11 = 2 \times 4 + \textcolor{#0099ff}{3}}
\latex{11 = 3 \times 3 + \textcolor{#0099ff}{2}}
\latex{11 = 5 \times 2 + \textcolor{#0099ff}{1}}
\latex{11 = 11 \times 1 + \textcolor{#0099ff}{0}}
If you choose four-seater kayaks, 3 people will be left out. The number of kayakers left out is two for the 3-person and one for the 2-person kayaks. Only single-person kayaks will do without leaving anybody out.
The remainder of \latex{ 11 } divided by \latex{ 4 } is \latex{ 3, } divided by \latex{ 3 } is \latex{ 2 }, divided by \latex{ 2 } is \latex{ 1 }, and divided by \latex{ 1 } is \latex{ 0 }.
Example 1
Zoe invited four of her classmates to her party. Her mother placed a tray of cookies on the table, and after each game, everyone took one. At the end of the last game, they realised there wouldn't be enough cookies for everyone. How many cookies were left on the tray if originally there had been at least 15 and at most 24 cookies? How many games did they play in total?
Solution
As five children are playing, the number of remaining cookies can be calculated by checking the remainder when the total is divided by 5.
\latex{15 = 3 \times 5 + \textcolor{#0099ff}{0}}
\latex{16 = 3 \times 5 + \textcolor{#0099ff}{1}}
\latex{17 = 3 \times 5 + \textcolor{#0099ff}{2}}
\latex{18 = 3 \times 5 + \textcolor{#0099ff}{3}}
\latex{19 = 3 \times 5 + \textcolor{#0099ff}{4}}
\latex{20 = 4 \times 5 + \textcolor{#0099ff}{0}}
\latex{21 = 4 \times 5 + \textcolor{#0099ff}{1}}
\latex{22 = 4 \times 5 + \textcolor{#0099ff}{2}}
\latex{23 = 4 \times 5 + \textcolor{#0099ff}{3}}
\latex{24 = 4 \times 5 + \textcolor{#0099ff}{4}}
You can see that when you divide by 5, the remainders are 0, 1, 2, 3 and 4. There could be 0, 1, 2, 3 or cookies left. From the multiples of 5, you can deduce that they played 3 or 4 games.
Natural numbers can be grouped according to their remainders when divided by 5. Numbers with the same remainder belong to the same group. For example, the numbers 2, 7, 12, 17, 22, 27, ... all belong to the same group because their remainder is 2 when divided by 5.
Natural numbers can also be grouped according to their remainders when divided by other numbers, for example, 2, 3, 4,...
remainder when divided by \latex{ 2 }
remainder when divided by \latex{ 3 }
remainder when divided by \latex{ 4 }
remainder when divided by \latex{ 5 }
Analyse the remainders of \latex{ 59 } and \latex{ 69 } when divided by \latex{ 2, 3, 4, } and \latex{ 5 }.
remainder when divided by | \latex{ 59 } | \latex{ 69 } |
\latex{ 2 } | \latex{ 1 }, since \latex{59 = 29 \times 2 + \textcolor{#0099ff}{1}}; | \latex{ 1 }, since \latex{69 = 34 \times 2 + \textcolor{#0099ff}{1}}; |
\latex{ 3 } | \latex{ 2 }, since \latex{59 = 19 \times 3 + \textcolor{#0099ff}{2}}; | \latex{ 0 }, since \latex{69 = 23 \times 3 + \textcolor{#0099ff}{0}}; |
\latex{ 4 } | \latex{ 3 }, since \latex{59 = 14 \times 4 + \textcolor{#0099ff}{3}}; | \latex{ 1 }, since \latex{69 = 17 \times 4 + \textcolor{#0099ff}{1}}; |
\latex{ 5 } | \latex{4 }, since \latex{59 = 11 \times 5 + \textcolor{#0099ff}{4}}; | \latex{ 4 }, since \latex{69 = 13 \times 5 + \textcolor{#0099ff}{4}}. |
The remainder is always smaller than the divisor.
The number of possible remainders equals the number of divisors.
Exercises
{{exercise_number}}. In PE lessons, the students warm up with a complex exercise consisting of 8 steps. If the sequence of steps is repeated, which posture is the 39th? Draw it in your notebook.
\latex{ 1. }\latex{ 2. }\latex{ 3. }\latex{ 4. }\latex{ 5. }\latex{ 6. }\latex{ 7. }\latex{ 8. }{{exercise_number}}. List the natural numbers less than \latex{ 70 } that have:
- a remainder of \latex{ 2 } when divided by \latex{ 3 };
- a remainder of \latex{ 1 } when divided by \latex{ 2 };
- a remainder of \latex{ 4 } when divided by \latex{ 6 };
- a remainder of \latex{ 8 } when divided by \latex{ 7 }.
{{exercise_number}}. What are some of the common properties of the numbers
- at the top of the waves;
- at the bottom of the waves?
\latex{ 1 }\latex{ 3 }\latex{ 5 }\latex{ 7 }\latex{ 9 }\latex{ 11}\latex{ 12}\latex{ 10}\latex{ 8}\latex{ 6}\latex{ 4}\latex{ 2}\latex{ 0}\latex{ ... }{{exercise_number}}. How many of two adjacent natural number pairs are divisible by 2?
{{exercise_number}}. Before a game of hide-and-seek, the children decide who will be 'it' using a counting rhyme. At which child should the rhyme begin so that the first child ends up as 'it' by the end of the song?
Eeny-meeny-miny-moe
Catch-a-tiger-by-the-toe
If-he-hollers,-let-him-go
Eeny-meeny-miny-moe.
Catch-a-tiger-by-the-toe
If-he-hollers,-let-him-go
Eeny-meeny-miny-moe.
a)b)c)\latex{ 1. }\latex{ 2. }\latex{ 3. }\latex{ 4. }\latex{ 4. }\latex{ 1. }\latex{ 2. }\latex{ 3. }\latex{ 5. }\latex{ 1. }\latex{ 2. }\latex{ 3. }\latex{ 4. }\latex{ 5. }\latex{ 6. }{{exercise_number}}. What could be the remainder of a natural number when divided by \latex{ 7 }?
{{exercise_number}}. List the natural numbers less than \latex{ 100 } that have:
- a remainder of \latex{ 1 } when divided by \latex{ 7 };
- a remainder of \latex{ 5 } when divided by \latex{ 7 }.
{{exercise_number}}. How many possible remainders can a natural number have when divided by
- \latex{ 9; }
- \latex{ 14; }
- \latex{ 23}?
{{exercise_number}}. What property do the numbers of the same colour share?
\latex{ 1 }\latex{ 4 }\latex{ 7 }\latex{ 10 }\latex{ 11 }\latex{ 8 }\latex{ 5 }\latex{ 2 }\latex{ 0 }\latex{ 3 }\latex{ 6 }\latex{ 9 }\latex{ 12 }\latex{ ... }{{exercise_number}}. Out of three adjacent natural numbers, how many are divisible by 3?
{{exercise_number}}. Imagine a regular hexagon with the non-negative half of the number line 'wrapped' around it. Look at the remainders of the numbers at each vertex when divided by \latex{ 6 }. What do you notice?
\latex{ A }\latex{ B }\latex{ C }\latex{ D }\latex{ E }\latex{ F }\latex{ 24;18;12;6; }\latex{ 0 }\latex{ ;7;13;19 }\latex{ 1 }\latex{ ;8;14;20 }\latex{ 2 }\latex{ ;9;15;21 }\latex{ 3 }\latex{ 22;16;10; }\latex{ 4 }\latex{ 23;17;11; }\latex{ 5 }{{exercise_number}}. What shapes can be used to continue the following sequences, respectively?
a)b){{exercise_number}}. The remainder of \latex{ 39 }, when divided by \latex{ 6 }, is \latex{ 3 } because \latex{39 = 6 \times 6 + 3}. Find the remainders when \latex{ 39, 72, 85, 93, 100, 164, } and \latex{ 905 } are divided by:
- \latex{ 6; }
- \latex{ 7; }
- \latex{ 10. }
{{exercise_number}}. Identify the dividend, divisor, and remainder in the following expressions:
- \latex{8 \times 6 + 5};
- \latex{6 \times 13 + 12};
- \latex{5 \times 11 + 6};
- \latex{9 \times 7 + 0}.
{{exercise_number}}. By what rule are the lines and elements repeated? Which shape belongs to the following positions?
- the \latex{ 30 }th element in the \latex{ 15 }th row,
- the \latex{ 37 }th element in the \latex{ 17 }th row,
- the \latex{ 15 }th element in the \latex{ 23 }rd row,
- the \latex{ 13 }th element in the \latex{ 10 }th row.
\latex{ 1. }\latex{ 2. }\latex{ 3. }\latex{ 4. }\latex{ 5. }\latex{ .}\latex{ .}\latex{ .}{{exercise_number}}. Group the following numbers based on their remainders when divided by \latex{ 3 }:
\latex{ 29; 38; 75; 100; 207; 1,995; 2,006; 10,000 }.
How many groups are there? For each group, select any two numbers, calculate their sum and difference, and find the remainder when the result is divided by \latex{ 3 }.
Quiz
If Garfield wakes up hungry one day, he will wake up thirsty the next day, grumpy the next day, happy the next day, lazy the next day, and hungry again on the fifth day. How did he wake up on \latex{ 29 } September if he woke up hungry on \latex{ 1 } September?