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Rules of divisibility
Determining divisibility based on the last digit
The last digit of \latex{30}; \latex{140} and \latex{4,070} is \latex{0}. These numbers can be expressed as \latex{3\textcolor{00a1eb}{0}=3\times\textcolor{00a1eb}{10}}; \latex{14\textcolor{00a1eb}{0}=14\times\textcolor{00a1eb}{10}}; \latex{4,07\textcolor{00a1eb}{0}=407\times\textcolor{00a1eb}{10}}; therefore, they are divisible by \latex{ 10 }.
If the last digit of a natural number is \latex{0}, then it is divisible by \latex{10}.
If a natural number is divisible by \latex{ 10 }, it can be expressed as ten times a natural number. For example, \latex{5\times\textcolor{00a1eb}{10}=5\textcolor{00a1eb}{0}}; \latex{96\times\textcolor{00a1eb}{10}=96\textcolor{00a1eb}{0}}; \latex{230\times\textcolor{00a1eb}{10}=2,30\textcolor{00a1eb}{0}}. The last digit of any natural number multiplied by ten is \latex{ 0 }.
If a natural number is divisible by \latex{ 10 }, it can be expressed as ten times a natural number. For example, \latex{5\times\textcolor{00a1eb}{10}=5\textcolor{00a1eb}{0}}; \latex{96\times\textcolor{00a1eb}{10}=96\textcolor{00a1eb}{0}}; \latex{230\times\textcolor{00a1eb}{10}=2,30\textcolor{00a1eb}{0}}. The last digit of any natural number multiplied by ten is \latex{ 0 }.
If a natural number is divisible by \latex{10}, its last digit is \latex{0}.
The two rules combined:
A natural number is divisible by \latex{10} if and only if its last digit is \latex{0}.
Example 1
John goes for a walk. Which foot will he use to take the \latex{ 21 }st , the \latex{ 58 }th and the \latex{ 387 }th step if he takes the first step with his right foot?
Solution
John takes the \latex{ 1 }st, the \latex{ 3 }rd, the \latex{ 5 }th, the \latex{ 7 }th, the \latex{ 9 }th, ... step with his right foot. He takes the \latex{ 2 }nd, the \latex{ 4 }th, the \latex{ 6 }th, the \latex{ 8 }th, the \latex{ 10 }th, ... step with his left foot, including all the multiples of ten as well.
Every other step can be expressed as the sum of a multiple of \latex{ 10 } and a one-digit number.
\latex{21=20+1}
\latex{58=50+8}
\latex{387=380+7}
left
right
left
left
left
right
John takes the \latex{ 21 }st and \latex{ 387 }th steps with his right foot, while the \latex{ 58 }th step is taken with his left.
Numbers not divisible
by \latex{ 2 }: \latex{1; 3; 5; 7; 9; ...}
Numbers divisible by \latex{2}: \latex{0; 2; 4; 6; 8; 10; ...}
If the last digit of a natural number is \latex{0; 2; 4; 6} or \latex{8}, then it is divisible by \latex{2}.
If a natural number is divisible by \latex{2}, its last digit must be \latex{0; 2; 4; 6} or \latex{8}.
The two rules combined:
A natural number is divisible by \latex{2} if and only if its last digit is \latex{0; 2; 4; 6} or \latex{8}.
Example 2
Gina was reading a book and noticed that pictures were only on pages whose numbers were divisible by \latex{ 5 }. The book contains \latex{ 84 } pages. List all the pages that had pictures. What do you notice?
Solution
Pages with pictures: \latex{5;\; 10;\; 15;\; 20;\; 25;\; 30;\; 35;\; 40;}
\latex{ 45;\; 50;\; 55;\; 60;\; 65;\; 70;\; 75;\; 80. }
\latex{ 45;\; 50;\; 55;\; 60;\; 65;\; 70;\; 75;\; 80. }
Note that the numbers of pages that have pictures end with \latex{ 0 } or \latex{ 5 }.
Explain what you noticed in the example above.
Numbers that end with \latex{0} (\latex{10;\; 20;\; 30;\; 40;\; 50;\; 60; ...}) are divisible by \latex{ 10 }; thus, they are also divisible by \latex{ 5 }, a factor of \latex{ 10 }.
Numbers that end with \latex{ 5 } can be expressed as an addition with two addends:
E.g. \latex{15 = \textcolor{469fe3}{10}+ 5;} \latex{ 25 = \textcolor{469fe3}{20}+ 5; } \latex{ 35 = \textcolor{469fe3}{30}+ 5; } ... \latex{ 75 = \textcolor{469fe3}{70}+ 5. }
E.g. \latex{15 = \textcolor{469fe3}{10}+ 5;} \latex{ 25 = \textcolor{469fe3}{20}+ 5; } \latex{ 35 = \textcolor{469fe3}{30}+ 5; } ... \latex{ 75 = \textcolor{469fe3}{70}+ 5. }
Both addends are divisible by \latex{ 5 }. Thus, their sum is also divisible by \latex{ 5 }. Numbers that end with \latex{ 0 } or \latex{ 5 } are divisible by \latex{ 5 }.
If a number ends with \latex{ 1;\; 2;\; 3;\; 4;\; 6;\; 7;\; 8 } or \latex{ 9 }; it is not divisible by \latex{ 5 }.
\latex{10}
\latex{1\times10}
\latex{2\times5}
\latex{5} is a factor of \latex{10}.
Example:
\latex{31=3\times}
\latex{10}
\latex{+}
\latex{1}
\latex{32=3\times}
\latex{10}
\latex{+}
\latex{2}
\latex{33=3\times}
\latex{10}
\latex{+}
\latex{3}
\latex{34=3\times}
\latex{10}
\latex{+}
\latex{4}
divisible
by \latex{5}
by \latex{5}
\latex{36=3\times}
\latex{10}
\latex{+}
\latex{6}
\latex{37=3\times}
\latex{10}
\latex{+}
\latex{7}
\latex{38=3\times}
\latex{10}
\latex{+}
\latex{8}
\latex{39=3\times}
\latex{10}
\latex{+}
\latex{9}
not divisible
by \latex{5}
by \latex{5}
divisible
by \latex{5}
by \latex{5}
not divisible
by \latex{5}
by \latex{5}
\latex{1}
\latex{2}
\latex{3}
\latex{4}
\latex{1}
\latex{2}
\latex{3}
\latex{4}
remainder
remainder
A natural number is divisible by \latex{5} if and only if its last digit is \latex{0} or \latex{5}.
Determining divisibility based on the last two digits
A natural number is divisible by \latex{100} if and only if its last two digits are \latex{0}.
Example:
\latex{4\textcolor{469fe3}{00}=4\,\times}
\latex{7,0\textcolor{469fe3}{00}=70\,\times}
\latex{100}
\latex{100}
\latex{403\,\times}
\latex{100}
\latex{800\,\times}
\latex{100}
\latex{=40,3\textcolor{469fe3}{00}}
\latex{=80,0\textcolor{469fe3}{00}}
the last two
digits are \latex{0}
digits are \latex{0}
divisible
by \latex{100}
by \latex{100}
divisible
by \latex{100}
by \latex{100}
the last two
digits are \latex{0}
digits are \latex{0}
If one of the factors is \latex{ 100, } the product is divisible by \latex{ 100 } and by all the factors of \latex{ 100 }, including \latex{ 4; 25; 20 } and \latex{ 50 } as well.
\latex{100}
\latex{1\times100}
\latex{2\times50}
\latex{4\times25}
\latex{5\times20}
\latex{10\times10}
Example 3
Without performing the divisions, determine whether \latex{ 728;\ 5,812 } and \latex{ 5,821 } are divisible by \latex{ 4 }.
Solution
These numbers can be expressed as an addition with two addends:
\latex{7}
\latex{28}
\latex{=7}
\latex{\times100}
\latex{+}
\latex{28}
\latex{5,8}
\latex{12}
\latex{=58}
\latex{\times100}
\latex{+}
\latex{12}
\latex{5,8}
\latex{21}
\latex{=58}
\latex{\times100}
\latex{+}
\latex{21}
The product is divisible by \latex{100},
so it is also divisible by \latex{4}, a factor of \latex{100}.
so it is also divisible by \latex{4}, a factor of \latex{100}.
Observe the last
two digits.
two digits.
\latex{\longrightarrow}
\latex{\downarrow}
\latex{4} is a factor of \latex{28};
\latex{4} is a factor of \latex{12};
\latex{4} is not a factor of \latex{21}.
\latex{4} is a factor of \latex{12};
\latex{4} is not a factor of \latex{21}.
\latex{ 728 } and \latex{ 5,812 } are divisible by \latex{ 4 }, while \latex{ 5,821 } is not.
A natural number is divisible by \latex{4} if and only if the number formed by its last two digits is divisible by \latex{4}.
Similarly, when determining whether a natural number is divisible by \latex{ 20; 25 } or \latex{ 50 }, it is enough to focus on the last two digits.
For example, \latex{3,475} is divisible by \latex{25} because:
\latex{3,4}
\latex{75}
\latex{=34}
\latex{\times100}
\latex{+}
\latex{75}
The product is divisible by \latex{100},
so it is also divisible by \latex{25}, a factor of \latex{100}.
so it is also divisible by \latex{25}, a factor of \latex{100}.
The number formed by the last
two digits is divisible by \latex{25}.
two digits is divisible by \latex{25}.
A natural number is divisible by \latex{25} if and only if the number formed by its last two digits is divisible by \latex{25}.
When dividing a natural number by four, the remainder equals the remainder of the last two digits when divided by four.
E.g.
\latex{25,472=25,400+72} and \latex{72=18\times 4+\textcolor{009fe3}{0}}, the remainder of \latex{ 25,472 } when divided by \latex{ 4 } is \latex{ 0 };
\latex{25,469 = 25,400 + 69} and \latex{69=17\times 4+\textcolor{009fe3}{1}}, the remainder of \latex{ 25,469 } when divided by \latex{ 4 } is \latex{ 1 }.
\latex{25,469 = 25,400 + 69} and \latex{69=17\times 4+\textcolor{009fe3}{1}}, the remainder of \latex{ 25,469 } when divided by \latex{ 4 } is \latex{ 1 }.
Similarly, when dividing a natural number by \latex{ 20; 25; 50 } and \latex{ 100 }, the remainder equals the remainder of the last two digits when divided by these.
Determining divisibility based on the last three digits
A natural number is divisible by \latex{1,000} if and only if its last three digits are \latex{0}.
Example:
\latex{2,}
\latex{000}
\latex{2\times}
\latex{1,000}
\latex{4,500,}
\latex{000}
\latex{=4,500\times}
\latex{1,000}
\latex{90\times}
\latex{1,000}
\latex{=90,}
\latex{000}
\latex{1,803\times}
\latex{1,000}
\latex{=1,803,}
\latex{000}
the last \latex{3}
digits are \latex{0}
digits are \latex{0}
divisible
by \latex{1,000}
by \latex{1,000}
divisible
by \latex{1,000}
by \latex{1,000}
the last \latex{3}
digits are \latex{0}
digits are \latex{0}
If one of the factors is \latex{ 1,000 }, the product is divisible by \latex{ 1,000 } and by all the factors of
\latex{ 1,000 }, including \latex{ 8; 125; 250 } and \latex{ 500 } as well.
Example 4
Without performing the divisions, determine whether \latex{ 5,432;\ 17,128 } and \latex{ 7,324 } are divisible by \latex{ 8 }.
Solution
These numbers can be expressed as an addition with two addends:
\latex{5,}
\latex{432}
\latex{=\,5}
\latex{\times1,000}
\latex{+}
\latex{432}
\latex{17,}
\latex{128}
\latex{=17}
\latex{\times1,000}
\latex{+}
\latex{128}
\latex{5,}
\latex{324}
\latex{=58}
\latex{\times1,000}
\latex{+}
\latex{324}
The product is divisible by \latex{1,000},
so it is also divisible by \latex{8}, a factor of \latex{1,000}.
so it is also divisible by \latex{8}, a factor of \latex{1,000}.
Observe the last
three digits.
three digits.
\latex{432 = 54\times8} \latex{\rightarrow} \latex{8} is a factor of \latex{432};
\latex{128 = 16\times8} \latex{\rightarrow} \latex{8} is a factor of \latex{128};
\latex{324 = 40\times8+4} \latex{\rightarrow} \latex{8} is not a factor of \latex{324}.
\latex{128 = 16\times8} \latex{\rightarrow} \latex{8} is a factor of \latex{128};
\latex{324 = 40\times8+4} \latex{\rightarrow} \latex{8} is not a factor of \latex{324}.
\latex{ 5,432 } and \latex{ 17,128 } are divisible by \latex{ 8 }, while \latex{ 7,324 } is not.
\latex{1,000}
\latex{1\times 1,000}
\latex{2\times 500}
\latex{4\times 250}
\latex{5\times 200}
\latex{8\times 125}
\latex{10\times 100}
\latex{20\times 50}
\latex{25\times 40}
\latex{8} is a factor of \latex{1,000}.
A natural number is divisible by \latex{8} if and only if the number formed by its last three digits is divisible by \latex{8}.
Similarly, when determining whether a natural number is divisible by \latex{ 125; 200; 250 } or \latex{ 500 }, it is enough to focus on the last three digits.
For example, \latex{76,250} is divisible by \latex{125} because:
\latex{76,}
\latex{250}
\latex{=76}
\latex{\times1,000}
\latex{+}
\latex{250}
The product is divisible by 1,000,
so it is also divisible by 125
a factor of 1,000.
so it is also divisible by 125
a factor of 1,000.
The number formed by
the last three digits is
divisible by 125.
the last three digits is
divisible by 125.
A natural number is divisible by \latex{125} if and only if the number formed by its last three digits is divisible by \latex{125}.
When dividing a natural number by eight, the remainder equals the remainder of the last three digits when divided by eight.
E.g.
\latex{25,472 = 25,000 + 472} and \latex{472 = 59\times8+\textcolor{009fe3}{0}},
so the remainder of \latex{25,472} when divided by \latex{ 8 } is \latex{ 0 };
\latex{25,469 = 25,000 + 469} and \latex{469 = 58\times8+\textcolor{009fe3}{5}},
so the remainder of \latex{25,469} when divided by \latex{ 8 } is \latex{ 5 }.
Similarly, when dividing a natural number by \latex{ 40; 125; 200; 250; 500 } and \latex{ 1,000 }, the remainder equals the remainder of the last three digits when divided by these.
What is the remainder of \latex{276} and \latex{1,276} when divided by
- \latex{125};
- \latex{200};
- \latex{500?}
- \latex{276 = 2\times125+\textcolor{009fe3}{26}} \latex{1,276 = 1,000 + 276 = 8\times125+2\times125+\textcolor{009fe3}{26}}
- \latex{276 = 1\times200+\textcolor{009fe3}{76}} \latex{1,276 = 1,000 + 276 = 5\times200+1\times200+\textcolor{009fe3}{76}}
- \latex{276 = 0\times500+\textcolor{009fe3}{276}} \latex{1,276 = 1,000 + 276 = 2\times500+\textcolor{009fe3}{276}}
Showing the remainders on a number line:
\latex{0}
\latex{0}
\latex{0}
\latex{125}
\latex{250}
\latex{375}
\latex{500}
\latex{625}
\latex{750}
\latex{875}
\latex{1,000}
\latex{1,125}
\latex{1,250}
\latex{1,375}
\latex{1,500}
\latex{1,625}
\latex{200}
\latex{400}
\latex{600}
\latex{800}
\latex{1,000}
\latex{1,200}
\latex{1,400}
\latex{1,600}
\latex{500}
\latex{1,000}
\latex{1,500}
\latex{276}
\latex{1,276}
\latex{1,276}
\latex{1,276}
\latex{276}
\latex{276}
Exercises
{{exercise_number}}.
- Where would the set of numbers divisible by \latex{10,000} be in the diagram?
- Write true and false statements based on the diagram.
Divisible by \latex{10}
Divisible by \latex{100}
Divisible by \latex{1,000}
\latex{70}
\latex{11,230}
\latex{600}
\latex{3,400}
\latex{7,000}
\latex{82,000}
\latex{900,000}
\latex{1,000,000}
{{exercise_number}}.
- List two-digit numbers that can be the last two digits of numbers divisible by \latex{ 4 }.
- By which numbers can the numbers ending with \latex{ 000; 125; 250; 375; 500; 625; 750 } and \latex{ 875 } be divided?
{{exercise_number}}. List all the numbers divisible by \latex{125} which are not less than \latex{14,500}, and not greater than \latex{16,000}. How many such numbers are there?
{{exercise_number}}. List all the numbers divisible by \latex{ 25 } that are not less than \latex{ 570 } and not greater than \latex{ 850 }.
{{exercise_number}}. Determine the remainder of \latex{783;\ 3,689;\ 4,592;\ 7,840;\ 11,999} when divided by
- \latex{2};
- \latex{4};
- \latex{5};
- \latex{8};
- \latex{25};
- \latex{125.}
{{exercise_number}}. What is the remainder of the sum of \latex{2,787 + 3,058 + 12,429} divided by
- \latex{2};
- \latex{4};
- \latex{5};
- \latex{25};
- \latex{125};
- \latex{8?}
{{exercise_number}}. Copy this diagram into your notebook and write the following numbers into the correct sets.
\latex{0; 17; 45; 72; 30; 85; 160; 449; 328; 135; 794; 225; 900}.
What properties do the numbers in the common part of the two sets have?
Universal set
Divisible by \latex{2}
Divisible by \latex{ 5}
{{exercise_number}}. Copy this diagram into your notebook and write the following numbers into the correct sets.
\latex{7,356; 8,300; 94,050; 3,024; 875; 4,445; 1,932; 15,000; 18; 74; 125; 70, 900; 94.}
What properties do the numbers in the common part of the two sets have?
Universal set
Divisible by \latex{4}
Divisible by \latex{25}
{{exercise_number}}. Copy this diagram into your notebook and write the following numbers into the correct sets.
\latex{4,728; 152; 64; 1,250; 6,112; 415; 0; 94,375; 17,000; 500; 63,056; 16,875; 230,000.}
What properties do the numbers in the common part of the two sets have?
Universal set
Divisible by \latex{8}
Divisible by \latex{125}
{{exercise_number}}. Draw a diagram with a set of numbers divisible by \latex{ 4 } and another with numbers divisible by \latex{ 8 }. Write the following numbers into the correct sets. What do you notice?
\latex{56; 20; 100; 172; 256; 7,344; 9,040; 13,912; 25,000; 528; 403,000; 1; 680,516; 0.}
{{exercise_number}}. Fill in the missing digits of the following numbers so that they are divisible by both \latex{ 4 } and \latex{ 5 }. What can you say about numbers divisible by both \latex{ 4 } and \latex{ 5? }
- \latex{3\triangle 0} \latex{ \triangle =}
- \latex{7\square 85} \latex{\square =}
{{exercise_number}}. Use the number cards \latex{0} , \latex{3} , \latex{4} and \latex{5} to create all the possible three-digit numbers. Make a diagram with a set for numbers divisible by \latex{ 2 } and another for those divisible by \latex{ 5 }. Write the numbers into the correct sets.
{{exercise_number}}. Use the number cards \latex{0} \latex{1} \latex{2} \latex{5} to create
- three-digit numbers divisible by \latex{ 4 }. Which of these are also divisible by \latex{ 5 }? What can you say about numbers divisible by both \latex{ 4 } and \latex{ 5 ?}
- three-digit numbers divisible by \latex{ 5; 25 } and \latex{ 50 }. Represent them in a diagram. Write true statements based on the diagram.
{{exercise_number}}. Decide whether the following statements are true or false.
- If the number formed by the last two digits of a number is divisible by \latex{ 4 }, then the number is also divisible by \latex{ 4 }.
- A number is divisible by \latex{4} if its last two digits are divisible by \latex{4.}
- If a number is divisible by both \latex{ 4 } and \latex{ 5 }, it is also divisible by \latex{ 20 }.
- If a number is divisible by both \latex{ 4 } and \latex{ 2 }, it is also divisible by \latex{ 8 }.
{{exercise_number}}. Decide whether the following statements are true or false.
- If a number is divisible by \latex{ 50 }, it is also divisible by \latex{ 5 }.
- Every number divisible by \latex{25} is also divisible by \latex{50}.
- If a number is a multiple of \latex{ 25 }, it is also a multiple of \latex{ 5 }.
- There is a number divisible by \latex{25} whose digits are all odd.
{{exercise_number}}. The organisers of a national maths competition told the participants what the first prize was using the following riddle: "Count from left to right by \latex{ 1 } in the following way: \latex{ 1.A, 2.B, 3.C, 4.D, 5.E, 6.D, 7.C, 8.B, 9.A, 10.B, 11.C, } ... The \latex{ 1,000 }th term of the sequence will show you the first prize." What is the first prize?
\latex{A)}
\latex{B)}
\latex{C)}
\latex{D)}
\latex{E)}
Quiz
A certain number of consecutive pages have fallen out of a book. The first missing page was page \latex{ 143 }, while the last one's page number included the same digits but in a different order. How many pages have fallen out of the book?
