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Ege, Burcu, Bana and Alara all sent in
examples of numbers that they had experimented with. They found
that their digits always added up to a multiple of nine, and that
the numbers themselves were also divisible by nine. They concluded
that there is a link between these two properties, so that if a
number has digits that sum up to nine, it must be a multiple of
nine. Here are some of their examples:
From the set of numbers from $1$ to $9$ and by using each number
once and once only;
Example 1 : $345 + 6789 + 210 = 7344$
=816*9
$7 + 3 + 4 + 4 = 18$. $18$ is a multiple of $9$, so the sum is
divisible by nine.
Example 2 : $1023 + 4 + 5 + 6 + 7 + 8 + 9 = 1062$
= 118*9
$1 + 0 + 6 + 2 = 9$. $9$ is a multiple of $9$, so the sum is
divisible by nine.
Example 3: $1234 + 56 + 789 = 2079$
=
231*9
$2 + 0 + 7 + 9 = 18$. $18$ is a multiple of $9$, so the sum is
divisible by nine.
Example 4: $6723 + 14589 = 21312$
=
2368*9
$2 + 1 + 3 + 1 + 2 = 9$. $9$ is a multiple of $9$, so the sum is
divisible by nine.
They also repeated the exercise for the set of
numbers $1-8$ and found that the result was the same:
Example 1 : $23 + 467 + 158 = 648$ =
72*9
$6 + 4 + 8 = 18$. 18 is a multiple of $9$, so the sum is
divisible by nine.
Example 2: $123 + 45 + 67 + 8 = 243$
=
27*9
$2 + 4 + 3 = 9$. $9$ is a multiple of $9$, so the sum is
divisible by nine.
Example 3: $6245 + 137 + 8 = 6390$
=
710*9
6 + 3 + 9 + 0 = 18. 18 is a multiple of 9, so the sum is
divisible by nine.
Example 4: $154 + 786 + 32 = 972$
=
108*9
$9 + 7 + 2 = 18$. $18$ is a multiple of $9$, so the sum is
divisible by nine.
And for the set of numbers $0-9$:
Example 1 : $1023 + 45 + 67 + 89 = 5679$
=
631*9
$5 + 6 + 7 + 9 = 27$. $27$ is a multiple of $9$, so the sum is
divisible by nine.
Rohaan from Longbay Primary School explained
why this always works for the sum of any numbers made from the
digits $1-9$:
I think the reason behind this is when you add all the digits
(from $1$ to $9$) the total is $45$. $45$ is divisible $9$ so
whatever groups of numbers you make and add up must be divisible by
$9$.
That's right, and the numbers $1-8$ add up to
$36$, which is also a multiple of $9$, so the rule still works. For
the sets of numbers $1-6$ and $1-5$, Ege and Banu found a similarly
interesting result for multiples of $3$:
Example 1 : $231 + 4 + 65 = 300$
=
100*3
$3 + 0 + 0 = 3$. $3$ is a multiple of $3$, so the sum is divisible
by three.
Example 2 : $12 + 34 + 56 = 102$
=
34*3
$1 + 0 + 2 = 3$. $3$ is a multiple of $3$, so the sum is divisible
by three.
So there you have it! This rule only
works for multiples of $3$ or $9$, but it makes it very quick and
easy to find out whether or not a big number is divisible by $3$ or
$9$ without using a calculator. Thank you for all your excellent
solutions.