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Mathematics 6.

Table of contents
Divisibility
The multiples and factors of natural numbers
Look at the remainder
The divisibility of sums, differences and products
Rules of divisibility
Determining divisibility based on the sum of the digits
Additional divisibility rules
Primes and composite numbers
Writing composite numbers as the product of prime numbers
Common factors, the highest common factor
Common multiples, lowest common multiple
Mixed exercises
How to solve problems?
Rational numbers I.
Axial symmetry
Rational numbers II.
Proportion
Percentages
Probability, statistics
Supplementary material
The multiples and factors of natural numbers
\latex{ 16 }
\latex{ 1 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ 6 }
\latex{ 7 }
\latex{ 8 }
\latex{ 9 }
\latex{ 10 }
\latex{ 11 }
\latex{ 12 }
\latex{ 13 }
\latex{ 14 }
\latex{ 15 }
Sphaera stellarum fixarum
Terra
Solis
Ignis
Lunæ
Mercurii
Veneris
Martis
Saturni
Iovis
Pythagorean World Model
Aer
Ancient Greek mathematicians were the first to deal with number theory, a branch of mathematics that studies the properties of numbers. In the \latex{ 6 }th century BC, Pythagoras and his disciples believed that if they could understand the relationship between numbers and their behaviour, they could reveal the secrets of the Universe. They claimed that the building block of the world was the number \latex{ 1 }, a unit that can be used to create all the other numbers.
If the pebbles representing a number can be arranged in the shape of a rectangle, the number is called a pronic or rectangular number. For example, \latex{ 6, 8, 12,... }etc. are pronic numbers.
\latex{6}
\latex{8}
\latex{12}
\latex{12}
\latex{2\times3}
\latex{2\times4}
\latex{2\times6}
\latex{3\times4}
\latex{=}
If the pebbles cannot be arranged to form a rectangle, the number is called a linear number. For example, \latex{ 2, 3, 5, 7, ... } etc. are linear numbers.
\latex{2}
\latex{3}
\latex{5}
\latex{7}
Square numbers, such as \latex{ 4, 9, 16,... } are special pronic numbers. In the case of square numbers, the pebbles can also be arranged in the shape of a square. Interestingly, the difference of neighbouring square numbers is always an odd number, and starting from \latex{ 1 }, the sum of odd numbers is always a square number.
\latex{1}
\latex{4}
\latex{9}
\latex{16}
\latex{25}
\latex{1+3}
\latex{1+3+5}
\latex{1+3+5+7}
\latex{1+3+5+7+9}
The relationship between the number of pebbles that can be arranged in the shape of a triangle (triangular numbers) is also worth examining: starting from \latex{ 1 }, the sum of natural numbers is a triangular number.
\latex{1}
\latex{3}
\latex{1+2}
\latex{6}
\latex{1+2+3}
\latex{10}
\latex{1+2+3+4}
\latex{15}
\latex{1+2+3+4+5}
Number theory studies integers (whole numbers). Now, you will learn about the relationships between natural numbers and perform calculations in the decimal system.
Natural numbers are \latex{ 0, 1, 2, 3, 4, 5, 6, 7, ... }
\latex{ {\displaystyle \mathbb {N} } = \left\{0;1;2;3,...\right\}}
The multiples of natural numbers
Example 1
In a board game, the animal figures can move according to special rules. Every piece starts from the 'Start' field. The sparrow jumps one field at a time; the frog jumps two fields; the rabbit, three; and the kangaroo, five. What fields can the animals move to?
Solution 
\latex{0\times1}
\latex{1\times1}
\latex{2\times1}
\latex{3\times1}
\latex{5\times1}
\latex{4\times1}
\latex{6\times1}
\latex{7\times1}
\latex{8\times1}
\latex{9\times1}
\latex{10\times1}
\latex{11\times1}
\latex{12\times1}
\latex{13\times1}
\latex{14\times1}
START
\latex{7 }
\latex{ 1 }
\latex{ 2 }
\latex{3 }
\latex{13 }
\latex{5 }
\latex{4 }
\latex{6 }
\latex{9}
\latex{8 }
\latex{14 }
\latex{11 }
\latex{10 }
\latex{12 }
The sparrow can jump on every natural number.
The multiples of \latex{ 1 } are:\latex{ 0; 1; 2; 3; 4; 5; ... }
with j jumps, the sparrow moves to field \latex{j\times1}
\latex{0\times2}
\latex{1\times2}
\latex{2\times2}
\latex{3\times2}
\latex{4\times2}
\latex{5\times2}
\latex{6\times2}
\latex{7\times2}
START
\latex{13 }
\latex{ 14 }
\latex{ 3 }
\latex{5 }
\latex{7 }
\latex{9 }
\latex{11 }
\latex{ 1 }
\latex{ 2 }
\latex{ 4 }
\latex{ 6 }
\latex{ 8 }
\latex{ 10 }
\latex{ 12 }
The frog can jump only on the multiples of \latex{ 2 }.
The multiples of \latex{ 2 } are: \latex{ 0; 2; 4; 6; 8; 10; ... }
with k jumps, the frog moves to field \latex{k\times2}
\latex{0\times3}
\latex{1\times3}
\latex{2\times3}
\latex{3\times3}
\latex{4\times3}
START
\latex{ 14 }
\latex{ 1 }
\latex{ 12 }
\latex{ 4 }
\latex{ 5 }
\latex{ 7 }
\latex{ 8 }
\latex{ 10 }
\latex{ 11 }
\latex{ 13 }
\latex{ 2 }
\latex{ 3 }
\latex{ 6 }
\latex{ 9 }
The rabbit can jump only on the multiples of \latex{ 3 }.
The multiples of \latex{ 3 } are: \latex{ 0; 3; 6; 9; 12; ... }
with m jumps, the rabbit reaches field \latex{m\times3}
\latex{0\times5}
\latex{1\times5}
\latex{2\times5}
START
\latex{14 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 10 }
\latex{6 }
\latex{7 }
\latex{8 }
\latex{9 }
\latex{11 }
\latex{12 }
\latex{13 }
\latex{ 4 }
\latex{ 5 }
The kangaroo can jump only on the multiples of \latex{ 5 }.
The multiples of \latex{ 5 } are: \latex{ 0; 5; 10; 15; 20; ... }
with n jumps, the kangaroo moves to field \latex{n\times5}
You can come to the following conclusions:
  • all natural numbers are multiples of \latex{ 1 };
  • every natural number is a multiple of itself;
  • every natural number different than \latex{ 0 } has an infinite number of multiples;
  • \latex{ 0 } is a multiple of every natural number.
673619
The factors of natural numbers
Example 2
Flowers are packed at a garden centre. The gardener would like to fit \latex{ 18 } cut flowers into a perforated cardboard box, arranged in rows so that there are the same number of flowers in each row. How can the flowers be arranged?
factors
\latex{1\times18=18}
\latex{2\times9=18}
\latex{3\times6=18}
\latex{\textcolor{#ff0000}{2}}
\latex{\textcolor{#ff0000}{9}}
\latex{\textcolor{#619FCF}{18}}
\latex{\textcolor{#619FCF}{1}}
\latex{\textcolor{#009E72}{3}}
\latex{\textcolor{#009E72}{6}}
the factors of \latex{ 18 }:\latex{ 1; 2; 3; 6; 9; 18. }
Solution 
Find pairs of numbers whose product is \latex{ 18 }.
\latex{1\times18=18;}
\latex{18\times1=18;}
\latex{2\times9=18;}
\latex{9\times2=18;}
\latex{3\times6=18;}
\latex{6\times3=18.}
\latex{ 18 } is a multiple of \latex{ 1 } and \latex{ 18, 2 } and \latex{ 9, 3 } and \latex{ 6 }; that is, \latex{ 1, 2, 3, 6, 9, } and \latex{ 18 } are factors of \latex{ 18. }
The flowers can be packed in \latex{ 1, 2, 3, 6, 9, } or \latex{ 18 } rows.
You can list every factor of a number by finding factor pairs.
It is possible to determine whether any natural number is a factor of any other natural number.
Example 3
Is the first number a factor of the second one?
  1. \latex{ 6 } of \latex{ 24 }
  1. \latex{ 17 } of \latex{ 51 }
  1. \latex{ 0 } of \latex{ 0 }
  1. \latex{ 6 } of \latex{ 10 }
\latex{ 6 } is a factor of \latex{ 24 };
\latex{ 24 } is a multiple of \latex{ 6 };
\latex{ 24 } is divisible by \latex{ 6 }.
Solution 
  1. \latex{ 6 } is a factor of \latex{ 24 } because there is a natural number (\latex{ 4 }), which multiplied by \latex{ 6 } is 24: \latex{4 \times6 = 24}.
  2. \latex{ 17 } is a factor of \latex{ 51 } because there is a natural number (\latex{ 3 }), which multiplied by \latex{ 17 } is\latex{ 51 }: \latex{3\times17=51}.
  3. \latex{ 0 } is a factor of \latex{ 0 } because there is a natural number, for example, \latex{ 5 }, which multiplied by \latex{ 0} is \latex{ 0:} \latex{5\times0=0}.
  4. \latex{ 6 } is not a factor of \latex{ 10 } because no natural number multiplied by \latex{ 6 } is \latex{ 10 }.
\latex{ 6 } is not a factor of \latex{ 10 };
\latex{ 10 } is not a multiple
of \latex{ 6 };
\latex{ 10 } is not divisible
by \latex{ 6 }.
Natural number \latex{ a } is a factor of natural number \latex{ b } if there is a natural number \latex{ k }, which multiplied by a gives \latex{ b }; that is \latex{k \times a = b}.
\latex{ a } is a factor of \latex{ b };
\latex{ b } is a multiple of \latex{ a };
\latex{ b } is divisible by \latex{ a }.
\latex{ 1 } is a factor of every natural number.
Every natural number is a factor of itself.
Every natural number is a factor of zero.
The trivial factors of a number are \latex{ 1 } and the number itself.
For example, the proper factors of \latex{ 15 } are \latex{ 3 } and \latex{ 5 }, while its trivial factors are \latex{ 1 } and \latex{ 15 }.
\latex{0\times n = 0},
where \latex{n = 0; 1; 2};
...
673621
Exercises
{{exercise_number}}. What distance was covered by a snow groomer if its caterpillar tracks are \latex{ 4 \,m} long and they completed \latex{ 2, 3 } and \latex{ 4 } turns?
{{exercise_number}}. Write down the ten smallest positive multiples of \latex{ 17 }.
{{exercise_number}}. A person must take one pill every two hours. How long does it take the patient to take \latex{ 6 } pills according to the doctor's instructions?
{{exercise_number}}. You want to saw planks to pieces of equal length. It takes you \latex{ 3 } minutes to saw a plank into \latex{ 4 } pieces. How long does it take you to saw \latex{ 5 } planks in \latex{ 4 } pieces?
{{exercise_number}}. Which of the numbers \latex{ 2, 3, 4, 6, 8, } and \latex{ 9 } are factors of \latex{ 12 }? Why?
{{exercise_number}}.  Which of the numbers \latex{ 2, 3, 4, 7, 17, } and \latex{ 34 } are factors of \latex{ 34 }? Why?
{{exercise_number}}. Mark the factors of \latex{ 12 } with red on a number line and those of \latex{ 24 } with blue. What do you notice?
{{exercise_number}}. Which of the numbers \latex{ 0, 9, 46, 54, 99, } and \latex{ 109 } are multiples of \latex{ 9 }?
{{exercise_number}}. In a supermarket, the temperature of the refrigerators is measured when the store opens and every \latex{ 90 } minutes after that. If the store is open for \latex{ 12 } hours, at what minutes after opening is the temperature of the refrigerators measured?
{{exercise_number}}. Which multiples of 16 are not smaller than 160 and not greater than 304?
{{exercise_number}}. A bus leaves the station every 8 minutes. List every minute when a bus leaves the station during the following two hours.
{{exercise_number}}. Which numbers not smaller than \latex{ 210 } and not larger than \latex{ 325 } are the multiples of \latex{ 21 }?
{{exercise_number}}. Write down the first ten multiples of \latex{ 7 } and \latex{ 8 }. Show the multiples in a Venn diagram. Which numbers are found in the common set?
{{exercise_number}}. Write down the ten smallest positive multiples of \latex{ 9 } and \latex{ 15 }. Show the multiples in a Venn diagram. Which multiples are found in the common set? Which numbers are the multiples of both \latex{ 9 } and \latex{ 15 }? Which number is the smallest among the common multiples?
{{exercise_number}}. List the factors of 16 and 18. Make a Venn diagram, containing all of them. Which are the common factors of 16 and 18?
{{exercise_number}}.
  1. List the factors of \latex{ 25 }.
  2. List the factors of \latex{ 20 }.
  3. Circle the numbers that are factors of both numbers. How many common factors do they have?
{{exercise_number}}. How many pages can books contain if they are cut from a whole number of parent sheets, each divided into 16 pages?
{{exercise_number}}. Where is cabin 5 of the big wheel found after turning \latex{ 180º, 360º, 720º } and \latex{ 900º }?
\latex{ 7 }
\latex{ 1 }
\latex{ 5 }
\latex{ 4 }
\latex{ 6 }
\latex{ 8 }
\latex{ 2 }
\latex{ 3 }
{{exercise_number}}. List the multiples of \latex{ 13 } that are not larger than \latex{ 150 }. Write true statements about these numbers.
{{exercise_number}}. Four 1-metre-long ribbons are tied together. How many centimetres shorter will the resulting ribbon be if a 7 cm section of each ribbon is needed to tie them together?
{{exercise_number}}. Two squirrels share 30 acrons. One of them takes two acorns for every three the other took. How many acorns does each get?
{{exercise_number}}. Make a Venn diagram, including the factors of \latex{ 18 } and \latex{ 36 }. Is there an empty set? If yes, make a Venn diagram without an empty set.
{{exercise_number}}. Find natural numbers that
  1. do not have any factors;
  2. have exactly one factor;
  3. have exactly two factors;
  4. have exactly four factors.
{{exercise_number}}. Based on the pattern, write the factors of \latex{ 20 } and \latex{ 36 } into the circles. Every arrow must point towards a multiple.
\latex{ 16 }
\latex{ 8 }
\latex{ 4 }
\latex{ 2 }
\latex{ 1 }
\latex{ 1 }
\latex{ 7 }
\latex{ 35 }
\latex{ 5 }
\latex{ 1 }
\latex{ 20 }
\latex{ 1 }
\latex{ 36 }
Quiz
Find numbers that have an odd number of factors.

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