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lfdahl
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Find all integers, $n$, such that the set $\{1,2,3,4, ...,n\}$ can be written as the disjoint union of the subsets, $A$, $B$ and $C$ -whose sums of elements are equal.
lfdahl said:Find all integers, $n$, such that the set $\{1,2,3,4, ...,n\}$ can be written as the disjoint union of the subsets, $A$, $B$ and $C$ -whose sums of elements are equal.
Albert said:$S_n=\dfrac {n(n+1)}{2}$ must be a multiple of 3, and $n>4$
if :$n=5,$$S_5=15$ and $\dfrac {15}{3}=5=$ sums of elements
we may set :{$A=1,4$},{$B=2,3$}, {$C=5$} and all its combinations
if :$n=6,$$S_6=21$ and $\dfrac {21}{3}=7=$ sums of elements
we may set :{$A=1,6$},{$B=3,4$}, {$C=2,5$} and all its combinations
from above we get :
case 1: $n=5,8,11,14,---=3p+2 ,p\geq 1$
case 2: $n=6,9,12,15,---=3q+3 ,q\geq 1$
lfdahl said:Hi, kaliprasad and Albert. Thankyou for your participation and clever solutions!
Your discussion on the possible values of $n$ is interesting, and the suggested solution below uses a surprisingly short and clear argument:
To find the possible $n$-values, observe that:$\sum_{x \in A}+\sum_{x \in B}+\sum_{x \in C} = \frac{1}{2}n(n+1)$, which is divisible by $3$.It follows, that $n$ must be congruent to $0,2,3$ or $5$ modulo $6$. Therefore $n \le 4$ can be excluded.So the list of $n$-values begins with: $n \in \left\{5,6,8,9,11, ...\right\}$For $n = 5,6,8,9$ we have the following partitions:\[ n = 5:\: \: \: \: A = \left \{ 1,4 \right \}\: \: \: B = \left \{ 2,3 \right \}\: \: \: C = \left \{ 5 \right \} \\\\ n = 6:\: \: \: \: A = \left \{ 1,6 \right \}\: \: \: B = \left \{2,5 \right \}\: \: \: C = \left \{3,4 \right \} \\\\ n = 8:\: \: \: \: A = \left \{1,2,3,6 \right \}\: \: \: B = \left \{ 5,7 \right \}\: \: \: C = \left \{ 4,8 \right \} \\\\ n = 9:\: \: \: \: A = \left \{ 1,2,3,4,5 \right \}\: \: \: B = \left \{7,8 \right \}\: \: \: C = \left \{ 6,9 \right \} \\\\ \]Now, to proceed with a $n+6$-value (e.g. $11$), we can use the nice procedure, which kaliprasad suggested:Use the partition depicted for $n=5$: Now join $n+1$ and $n+6$ to $A$, $n+2$ and $n+5$ to $B$ - and $n+3$ and $n+4$ to $C$.
Integers are whole numbers, both positive and negative, and zero. They do not include fractions or decimals.
To find all the integers that satisfy a given condition, you can use mathematical techniques such as algebra or number theory. Alternatively, you can use a computer program or calculator to generate a list of integers that fit the criteria.
The restrictions on the values of the integers will depend on the specific problem or condition given. Some problems may have a limited range of values, while others may allow for any integer to be a solution.
To ensure that you have found all the possible solutions, you can check your work by plugging the integers back into the original equation or condition. You can also use mathematical proofs or algorithms to verify that you have found all the possible solutions.
One example of a problem involving finding all integers is "Find all integers x such that 2x + 1 = 11." The solutions to this problem are x = 5 and x = 10, as plugging these values into the equation results in 11 on both sides.