MHB Solve for $\overline{abc}:$ $N=3194$

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The problem involves finding the three-digit number $\overline{abc}$ such that the sum of the permutations $\overline{acb}+\overline{bac}+\overline{bca}+\overline{cab}+\overline{cba}$ equals 3194. The total sum of these permutations is expressed as $222(a+b+c)$, which must be a multiple of 222 and lie between 3194 and 4194. The valid multiples of 222 in this range are 3330, 3552, 3774, and 3996, corresponding to potential values of $\overline{abc}$. The only solution that satisfies both the sum condition and the digit sum condition is $\overline{abc} = 358$, where $3+5+8 = 16$. Thus, the final answer is $\overline{abc} = 358$.
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if $N=\overline{acb}+\overline{bac}+\overline{bca}+\overline{cab}+\overline{cba}=3194$ is the sum of five 3-digit numbers
please find :$\overline{abc}$
(here $\overline{abc}$ is also a 3-digit number)
 
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Albert said:
if $N=\overline{acb}+\overline{bac}+\overline{bca}+\overline{cab}+\overline{cba}=3194$ is the sum of five 3-digit numbers
please find :$\overline{abc}$
(here $\overline{abc}$ is also a 3-digit number)
[sp]The sum of all six numbers is $(100 + 10 + 1)(2a+2b+2c) = 222(a+b+c)$, so it has to be a multiple of $222$. Also, it has to lie between $3194$ and $4194$. The only multiples of $222$ in that range are $$\begin{aligned} 3330\ &= 15\times 222 \text{, corresponding to } \overline{abc} = 3330 - 3194 = 136, \\ 3552\ &= 16\times 222 \text{, corresponding to } \overline{abc} = 3552 - 3194 =358, \\ 3774\ &= 17\times 222 \text{, corresponding to } \overline{abc} = 3774 - 3194 =580, \\ 3996\ &= 18\times 222 \text{, corresponding to } \overline{abc} = 3996 - 3194 = 802. \end{aligned}$$ Also, the sum $a+b+c$ must be equal to the corresponding multiple of $222$. The only case where this occurs is when $\boxed{\overline{abc} = 358} = 16\times 222$, with $3+5+8 = 16$.[/sp]
 
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Opalg said:
[sp]The sum of all six numbers is $(100 + 10 + 1)(2a+2b+2c) = 222(a+b+c)$, so it has to be a multiple of $222$. Also, it has to lie between $3194$ and $4194$. The only multiples of $222$ in that range are $$\begin{aligned} 3330\ &= 15\times 222 \text{, corresponding to } \overline{abc} = 3330 - 3194 = 136, \\ 3552\ &= 16\times 222 \text{, corresponding to } \overline{abc} = 3552 - 3194 =358, \\ 3774\ &= 17\times 222 \text{, corresponding to } \overline{abc} = 3774 - 3194 =580, \\ 3996\ &= 18\times 222 \text{, corresponding to } \overline{abc} = 3996 - 3194 = 136. \end{aligned}$$ Also, the sum $a+b+c$ must be equal to the corresponding multiple of $222$. The only case where this occurs is when $\boxed{\overline{abc} = 358} = 16\times 222$, with $3+5+8 = 16$.[/sp]
very nice solution!
 
Opalg said:
[sp]The sum of all six numbers is $(100 + 10 + 1)(2a+2b+2c) = 222(a+b+c)$, so it has to be a multiple of $222$. Also, it has to lie between $3194$ and $4194$. The only multiples of $222$ in that range are $$\begin{aligned} 3330\ &= 15\times 222 \text{, corresponding to } \overline{abc} = 3330 - 3194 = 136, \\ 3552\ &= 16\times 222 \text{, corresponding to } \overline{abc} = 3552 - 3194 =358, \\ 3774\ &= 17\times 222 \text{, corresponding to } \overline{abc} = 3774 - 3194 =580, \\ 3996\ &= 18\times 222 \text{, corresponding to } \overline{abc} = 3996 - 3194 = 802. \end{aligned}$$ Also, the sum $a+b+c$ must be equal to the corresponding multiple of $222$. The only case where this occurs is when $\boxed{\overline{abc} = 358} = 16\times 222$, with $3+5+8 = 16$.[/sp]

almost same method solved at

 
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