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Additional divisibility rules
\latex{ 13. }
\latex{ 1. }
\latex{ 2. }
\latex{ 3. }
\latex{ 4. }
\latex{ 5. }
\latex{ 6. }
\latex{ 7. }
\latex{ 8. }
\latex{ 9. }
\latex{ 10. }
\latex{ 11. }
\latex{ 12. }
Example 1
Kate put ice cubes in every second glass of lemonade and a straw in every third. Which glasses have both ice cubes and a straw in them if she makes drinks for \latex{ 19 } guests?
Solution
Write down the numbers of the glasses and mark the ones that contain ice cubes with blue.
\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};
\latex{16};
\latex{17};
\latex{18};
\latex{19}.
Numbers divisible by \latex{ 2 } are marked with blue.
Frame the numbers of the glasses containing straws.
\latex{1};
\latex{2};
\latex{\boxed{3;}}
\latex{4};
\latex{5};
\latex{\boxed{6;}}
\latex{7};
\latex{8};
\latex{\boxed{9;}}
\latex{10};
\latex{11};
\latex{\boxed{12;}}
\latex{13};
\latex{14};
\latex{\boxed{15;}}
\latex{16};
\latex{17};
\latex{\boxed{18;}}
\latex{19}.
Numbers divisible by \latex{ 3 } are framed.
Blue and framed numbers indicate the glasses containing ice cubes and a straw.
These are \latex{ 6,12 } and \latex{ 18 }, that is, numbers divisible by \latex{ 6 }.
Example 2
Write the following numbers in a diagram with two sets. One of the sets should contain numbers divisible by \latex{ 2 }, while the other those divisible by \latex{ 3 }.
\latex{0;\; 6;\; 8;\; 9;\; 10;\; 11;\; 12;\; 15;\; 18;\; 22;\; 24;\; 27;\; 33;\; 66;\; 98;\; 102;\; 107.}
Solution
Numbers divisible by both \latex{2} and \latex{3}, that is,
numbers divisible by \latex{6} as well
numbers divisible by \latex{6} as well
Universal set
Divisible by \latex{2}
Divisible by \latex{3}
\latex{11}
\latex{8}
\latex{10}
\latex{22}
\latex{0}
\latex{12}
\latex{18}
\latex{66}
\latex{24}
\latex{6}
\latex{102}
\latex{33}
\latex{9}
\latex{15}
\latex{27}
\latex{107}
A natural number is divisible by \latex{6} if and only if it is divisible by \latex{2} and \latex{3}.
Example 3
Write the numbers \latex{ 3;\; 12;\; 18;\; 16;\; 20;\; 24;\; 30;\; 32;\; 36;\; 40;\; 48 } and \latex{ 72 } in a diagram.
- The two sets should contain the natural numbers divisible by \latex{ 4 } and the natural numbers divisible by \latex{ 6 }.
- The two sets should contain the natural numbers divisible by \latex{ 3 } and the natural numbers divisible by \latex{ 8 }.
In which case will the intersection of the sets contain natural numbers divisible by
\latex{ 24 }?
Solution
Universal set
Divisible by \latex{4}
Divisible by \latex{6}
Divisible by \latex{3}
Divisible by \latex{8}
Universal set
Not all are divisible by \latex{24}.
Divisible by \latex{24}.
\latex{20}
\latex{32}
\latex{16}
\latex{40}
\latex{24}
\latex{48}
\latex{72}
\latex{36}
\latex{30}
\latex{18}
\latex{3}
\latex{36}
\latex{30}
\latex{3}
\latex{12}
\latex{24}
\latex{72}
\latex{48}
\latex{32}
\latex{40}
\latex{16}
\latex{20}
\latex{12}
A natural number is divisible by \latex{24} if and only if it is divisible by \latex{3} and \latex{8}.
Example 4
In the number \latex{7\triangle, 96\square} write digits in place of \latex{\triangle} and \latex{\square} so that the number would be divisible by both \latex{ 3 } and \latex{ 5 }.
Solution
A number is divisible by \latex{5} if its last digit is \latex{0} or \latex{5}. Thus, \latex{\square =0} or \latex{\square =5}.
A number is divisible by \latex{3} if the sum of its digits is divisible by \latex{3}.
If \latex{\square =0}, then the sum of the digits is
\latex{7+\triangle +9+6+0=22+\triangle .}
\latex{\triangle} can be replaced with \latex{0; 1; 2; ...; 9}, so the sum of the digits is between \latex{22} and \latex{31}. \latex{24; 27} and \latex{30} are divisible by \latex{3}, so \latex{\triangle} can be replaced with \latex{2; 5} or \latex{8}.
If \latex{\square =5}, then the sum of the digits is
\latex{7+\triangle +9+6+5=27+\triangle}.
The sum of the digits is between \latex{27} and \latex{36}. \latex{27;30; 33} and \latex{36} are divisible by \latex{3}, so \latex{\triangle}can be replaced with \latex{0; 3; 6} or \latex{9}.
The numbers you are looking for are: \latex{72,960;\ 75,960;\ 78,960;\ 70,965;\ 73,965;\ 76,965;\ 79,965}.
\latex{\triangle}
\latex{\square }
\latex{ 0 }
\latex{ 0 }
\latex{ 0 }
\latex{ 5 }
\latex{ 5 }
\latex{ 5 }
\latex{ 5 }
\latex{ 2 }
\latex{ 5 }
\latex{ 8 }
\latex{ 0 }
\latex{ 3 }
\latex{ 6 }
\latex{ 9 }
Overview
- Every natural number is divisible by \latex{ 1 }.
- Natural numbers ending in \latex{0}; \latex{2}; \latex{4}; \latex{6} and \latex{ 8 } are divisible by \latex{ 2 }.
- A natural number is divisible by \latex{ 3 } if the sum of its digits is divisible by \latex{ 3 }.
- A natural number is divisible by \latex{ 4 } if the number formed by its last two digits is divisible by \latex{ 4 }.
- Natural numbers ending in \latex{ 0 } or \latex{ 5 } are divisible by \latex{ 5 }.
- A natural number divisible by both \latex{ 2 } and \latex{ 3 } is also divisible by \latex{ 6 }.
- You have not learnt any divisibility rule for \latex{ 7 }.
- A natural number is divisible by \latex{ 8 } if the number formed by its last three digits is divisible by \latex{ 8 }.
- A natural number is divisible by \latex{ 9 } if the sum of its digits is divisible by \latex{ 9 }.
Exercises
{{exercise_number}}. Draw diagrams in your notebook, including the following sets. What numbers does their intersection contain?
- A={Numbers divisible by \latex{4} that are not greater than \latex{40}};
B ={Numbers divisible by \latex{5} that are not greater than \latex{40}};
- C={Numbers divisible by \latex{9} that are not greater than \latex{40}};
D={Numbers divisible by \latex{6} that are not greater than \latex{40}};
- E ={Numbers divisible by \latex{3} that are not greater than \latex{40}};
F ={Numbers divisible by \latex{4} that are not greater than \latex{40}};
- G={Numbers divisible by \latex{4} that are not greater than \latex{40}};
H={Numbers divisible by \latex{8} that are not greater than \latex{40}}.
{{exercise_number}}. Build the following sets in a diagram.
A = universal set;
B = {numbers divisible by \latex{3}};
C = {numbers divisible by \latex{5}};
D = {numbers divisible by \latex{4}}.
Write the numbers in the set diagram.
A ={\latex{0;\; 17;\; 9;\; 20;\; 25;\; 24;\; 28;\; 30;\; 36;\; 40;\; 48;\; 51;\; 55;\; 56;\; 60;\; 105;\; 180;\; 183}}
A ={\latex{0;\; 17;\; 9;\; 20;\; 25;\; 24;\; 28;\; 30;\; 36;\; 40;\; 48;\; 51;\; 55;\; 56;\; 60;\; 105;\; 180;\; 183}}
Determine the rules of divisibility for \latex{ 12}; \latex{15} and \latex{ 20 }.
{{exercise_number}}. Which of the following statements are true? Justify your answer.
- Every number divisible by \latex{ 15 } is also divisible by \latex{ 5 }.
- Every number divisible by \latex{ 25 } is also divisible by \latex{ 125 }.
- If a number is divisible by both \latex{ 2 } and \latex{ 3 }, it is also divisible by \latex{ 6 }.
- If a number is divisible by \latex{ 6 }, it must be even.
- If a number is divisible by \latex{ 12 }, it is also divisible by \latex{ 3 } and \latex{ 4 }.
- If a number is divisible by both \latex{ 6 } and \latex{ 8 }, it is also divisible by \latex{ 48 }.
{{exercise_number}}. Fill in the missing digits of the following numbers to make them divisible by both \latex{ 2 } and \latex{ 3 }.
a) \latex{9\square }
b) \latex{58\triangle }
c) \latex{1\heartsuit 7}
- \latex{4,7\circ\diamondsuit}
{{exercise_number}}. Fill in the missing digits of the following numbers to make them divisible by both \latex{ 3 } and \latex{ 4 }.
- \latex{\triangledown 24}
- \latex{58\square}
- \latex{\circ,63\triangle}
{{exercise_number}}. The digits of three-digit numbers are decided by throwing dice. Which of the following outcomes are certain, possible and impossible?
- The number you get will be smaller
than \latex{ 700 }.
- You will get an even number.
- You will get a number divisible by \latex{ 5 }.
- You will get a number divisible by \latex{ 10 }.
- You will get a number divisible by \latex{ 7 }.
- The number you get will be a multiple of \latex{ 1 }.
{{exercise_number}}. Fill in the missing digits so the resulting number is divisible by both \latex{ 3 } and \latex{ 5 }. Based on your results, make a table.
- \latex{89\square,53\triangle}
- \latex{7\circ4,25\heartsuit }
{{exercise_number}}. The digits of three-digit numbers are determined by throwing dice. How many three-digit numbers can you get that are divisible by
- \latex{3};
- \latex{4};
- \latex{6};
- \latex{15?}
{{exercise_number}}. The digits of three-digit numbers are determined by throwing dice. Is the difference of the largest and smallest possible numbers divisible by
- \latex{3};
- \latex{6?}
Justify your answer without performing the subtraction.
{{exercise_number}}. Copy the following diagrams into your notebook. Which diagram corresponds to the sets below? Label the sets and write some numbers in them.
I.
II.
III.
1.
2.
3.
A ={divisible by \latex{2}};
B ={divisible by \latex{4}};
C ={divisible by \latex{5}}.
D ={divisible by \latex{2}};
E ={divisible by \latex{3}};
F ={divisible by \latex{5}}.
G ={divisible by \latex{3}};
H ={divisible by \latex{9}};
I ={divisible by \latex{18}}.
Quiz
Are there more six-digit numbers divisible by \latex{ 3 } with a format of \latex{ababab} or six-digit numbers divisible by \latex{ 7 } with a format of \latex{abaaba?}
