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Determining divisibility based on the sum of the digits
The key opens the door whose
number is divisible by \latex{9}. You have
\latex{10} seconds to answer, or you will
remain my servant forever.
number is divisible by \latex{9}. You have
\latex{10} seconds to answer, or you will
remain my servant forever.
In a computer game, Frank arrives at the hall shown in the image above. He needs to open one of the doors to advance. Can he solve the task in such a short time?
Example 1
Use the following number cards to create three-digit numbers divisible by \latex{ 9 }. How many possible solutions are there?
- \latex{3} \latex{7} \latex{8}
- \latex{3} \latex{5} \latex{6}
Solution (a)
\latex{378 = 42\times9}
\latex{387 = 43\times9}
\latex{738 = 82\times9}
\latex{783 = 87\times9}
\latex{837 = 93\times9}
\latex{873 = 97\times9}
\latex{387 = 43\times9}
\latex{738 = 82\times9}
\latex{783 = 87\times9}
\latex{837 = 93\times9}
\latex{873 = 97\times9}
All of the three-digit numbers that can be created using number cards
\latex{3} , \latex{7} and \latex{8} are divisible
by \latex{9}.
Solution (b)
\latex{356 = 39\times9+\textcolor{00a1e4}{5}}
\latex{365 = 40\times9+\textcolor{00a1e4}{5}}
\latex{536 = 59\times9+\textcolor{00a1e4}{5}}
\latex{563 = 62\times9+\textcolor{00a1e4}{5}}
\latex{635 = 70\times9+\textcolor{00a1e4}{5}}
\latex{365 = 40\times9+\textcolor{00a1e4}{5}}
\latex{536 = 59\times9+\textcolor{00a1e4}{5}}
\latex{563 = 62\times9+\textcolor{00a1e4}{5}}
\latex{635 = 70\times9+\textcolor{00a1e4}{5}}
\latex{653 = 72\times9+\textcolor{00a1e4}{5}}
None of the three-digit numbers that can be created with number cards
\latex{3} , \latex{5} and \latex{6} are divisible
by \latex{9}.
Based on exercises a) and b), you can conclude that a number's divisibility by \latex{ 9 } depends only on its digits and not on their order. In exercise a), the sum of the digits is \latex{ 18 } (a multiple of \latex{ 9 }), while in exercise b), the sum of the digits is \latex{ 14 } (not divisible by \latex{ 9 }).
In the following examples, you will see that determining whether a number is divisible by \latex{ 9 } depends on the sum of its digits.
Example 2
Without performing the divisions, determine whether the following numbers are divisible by \latex{ 9 }.
- \latex{3,456}
- \latex{4,567}
Solution
- Express the number as an addition.
\latex{3,456 = 3,000 + 400 + 50 + 6}
Express the addends as a product in the following way.
\latex{3,456 = 3\times1,000+4\times100+5\times10+6}
Express the factors \latex{ 10; 100 } and \latex{ 1,000 } as an addition with two addends. One of the addends should be divisible by \latex{ 9 }. Then, perform the multiplications.
\latex{3,000 = 3\times1,000=3\times(999+1)=3\times\textcolor{00a1e4}{999}+\textcolor{dd005a}{3}\times1}
\latex{400=4\times\;\;\;100=4\times\;(99+1)\;=\;4\times\textcolor{00a1e4}{99}+\textcolor{dd005a}{4}\times1}
\latex{400=4\times\;\;\;100=4\times\;(99+1)\;=\;4\times\textcolor{00a1e4}{99}+\textcolor{dd005a}{4}\times1}
\latex{50 = 5\times\;\;\;\;\;10=5\times\;\;\;(9+1)=\;5\times\;\;\;\textcolor{00a1e4}{9}+\textcolor{dd005a}{5}\times1}
\latex{\underline{+\;6}=6\times\;\;\;\;\;\; 1=} \latex{ =6\times\;\; \textcolor{00a1e4}{0}\;+\underline{\textcolor{dd005a}{6}}\times 1}
\latex{3,456} \latex{18}
\latex{a\times(b+c)=}
\latex{=a\times b+a\times c}
\latex{3,456=\underbrace{3\times \textcolor{00a1e4}{999}+4\times \textcolor{00a1e4}{99}+5\times \textcolor{00a1e4}{9}}_{\text{ divisible by 9}} +\underbrace{\textcolor{dd005a}{3}+\textcolor{dd005a}{4}+\textcolor{dd005a}{5}+\textcolor{dd005a}{6}}_{\text{ the sum of the digits}}}
\latex{3 + 4 + 5 + 6 = 18}; \latex{ 18 } is divisible by \latex{ 9 }; therefore, \latex{ 3,456 } is also divisible by \latex{ 9 }.
- \latex{4,567=\underbrace{4\times \textcolor{00a1e4}{993}+5\times \textcolor{00a1e4}{99}+6\times \textcolor{00a1e4}{9}}_{\text{ divisible by 9}}+\underbrace{\textcolor{dd005a}{4}+\textcolor{dd005a}{5}+\textcolor{dd005a}{6}+\textcolor{dd005a}{7}}_{\text{ the sum of the digits}}}
\latex{4+5+6+7=22}.
\latex{ 22 } is not divisible by \latex{ 9 }; therefore, \latex{ 4,567 } is not divisible by \latex{ 9 } either.
A natural number is divisible by \latex{9} if and only if the sum of its digits is divisible by \latex{9}.
Act out the problem in the opening picture. Have someone from the class measure \latex{ 10 } \latex{ seconds }. How many students answered correctly?
Write other multi-digit numbers on the 'doors', and figure out who can get all the keys.
Is there a clever way of calculating the answers?
\latex{3,672\rightarrow \underbrace{3+6}+\underbrace{7+2}\rightarrow}
\latex{5,479\rightarrow \underbrace{5+4}+7+\underbrace{9}\rightarrow}
divisible by \latex{ 9 };
\latex{7} is not divisible by \latex{9}.
divisible by \latex{9}
divisible by \latex{9}
Example 3
Without performing the division, determine whether \latex{ 5,748 } is divisible by \latex{ 3 }.
Solution
Express \latex{ 5,748 } as an addition.
\latex{5,748 = 5,000 + 700 + 40 + 8}
Rewrite the addends as a multiplication.
\latex{5,748 = 5\times1,000+7\times100+4\times10+8}
Express the factors \latex{ 10; 100 } and \latex{ 1,000 } as an addition with two addends. One of the addends should be divisible by \latex{ 3 }. Then, perform the multiplications.
\latex{5,000 = 5\times1,000=5\times(999+1)=5\times\textcolor{00a1e4}{999}+\textcolor{dd005a}{5}\times1}
\latex{700=7\times\;\;\;100=7\times\;(99+1)=\;7\times\textcolor{00a1e4}{99}+\;\textcolor{dd005a}{7}\times1}
\latex{40 = 4\times\;\;\;\;\;10=4\times\;\;\;(9+1)=\;4\times\;\;\;\textcolor{00a1e4}{9}+\textcolor{dd005a}{4}\times1}
\latex{\underline{+\;8}=8\times\;\;\;\;\;\; 1=} \latex{ =8\times\;\;\; \textcolor{00a1e4}{0}\;+\underline{\textcolor{dd005a}{8}}\times 1}
\latex{5,748} \latex{24}
\latex{3} is a factor of \latex{9}.
\latex{5,748=\underbrace{5\times \textcolor{00a1e4}{999}+7\times \textcolor{00a1e4}{99}+4\times \textcolor{00a1e4}{9}}_{\text{ divisible by 3}}+\underbrace{\textcolor{dd005a}{5}+\textcolor{dd005a}{7}+\textcolor{dd005a}{4}+\textcolor{dd005a}{8}}_{\text{the sum of the digits}}}
\latex{5 + 7 + 4 + 8 = 24}.
\latex{ 24 } is divisible by \latex{ 3 }; thus, \latex{ 5,748 } is also divisible by \latex{ 3 }.
A natural number is divisible by \latex{3} if and only if the sum of its digits is divisible by \latex{3}.
Divisible by \latex{3}
Divisible by \latex{9}
\latex{12}
\latex{15}
\latex{18}
\latex{27}
\latex{36}
Exercises
{{exercise_number}}. Make a diagram with two sets: one for numbers divisible by \latex{ 3 } and the other for numbers divisible by \latex{ 9 }. Then, write the following numbers into the correct sets.
\latex{ 93; \,\;117;\,\; 459;\,\; 6,210;\,\; 6,573;\,\; 14,754;\,\; 49,893;\,\; 74,634;\,\; 86,283;\,\; 234,711;\,\; 576,495;\,\; 883,452 }
{{exercise_number}}. Three siblings together bought a present for their grandmother. They divided the price of the present equally among themselves. Which present could they possibly buy?
€\latex{ 2 }.\latex{ 25 }
€\latex{ 3 }.\latex{ 15 }
€\latex{ 5 }.\latex{ 35 }
€\latex{ 1 }.\latex{ 75 }
{{exercise_number}}. What digits can replace the symbols so that the number would be divisible by \latex{ 3 }?
- \latex{594,1\square 3}
- \latex{753,\triangle 48}
- \latex{47\circ,258}
{{exercise_number}}. Find all the number pairs that can replace the symbols to make the numbers divisible by \latex{ 9 }. Make a table based on your results.
- \latex{546,\square 7\triangle}
- \latex{8\circ6,\heartsuit 52}
- \latex{98\diamondsuit, 1\triangledown 8}
{{exercise_number}}. Determine the missing digits so that the numbers are
- divisible by \latex{2};
- divisible by \latex{3};
- divisible by \latex{4};
- divisible by \latex{5};
- divisible by \latex{9};
- divisible by \latex{25}.
- \latex{4\square};
- \latex{7\square};
- \latex{4\square6};
- \latex{93\square};
- \latex{67\square};
- \latex{7,\square49}.
{{exercise_number}}. Decide whether the following statements are true or false.
- Every number divisible by \latex{ 3 } is also divisible by \latex{ 9 }.
- If the remainder of a number when divided by \latex{ 9 } is \latex{ 2 }, its remainder when divided by \latex{ 3 } is also \latex{ 2 }.
- There is a positive whole number whose remainder is different when divided by \latex{ 3 } and \latex{ 9 }.
- Every number divisible by \latex{ 9 } is divisible by \latex{ 3 }.
- Not every number divisible by \latex{ 9 } is odd.
{{exercise_number}}. Draw a similar diagram in your notebook and write the positive whole numbers greater than \latex{ 2,000 } and not greater than \latex{ 2,020 } in the correct sets.
What are the common properties of the numbers in each set?
Universal set
Multiples of \latex{3}
Multiples of \latex{9}
{{exercise_number}}. A mind reader asked the people sitting in seats \latex{ 1; 2; 3; 4; 5; 6 } and \latex{ 7 } to think of a multi-digit number, multiply it by \latex{ 9 }, add the number of their seats, write the result on a piece of paper, and place the paper in his hat. Then, the mind reader told the audience that he would guess what number was written on each paper.
How does the trick of the mind reader work?
Quiz
A flock of starlings invaded the trees in your garden. One starling perched on each tree, and one was left without a tree. One tree was left without starlings when they occupied the trees in pairs. How many starlings were in the flock, and how many trees were in your garden?
