A three-digit $\overline{abc}$ number is called Ecuadorian if it meets the following conditions: $\bullet$ $\overline{abc}$ does not end in $0$. $\bullet$ $\overline{abc}$ is a multiple of $36$. $\bullet$ $\overline{abc} - \overline{cba}$ is positive and a multiple of $36$. Determine all the Ecuadorian numbers.
Problem
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Tags: number theory, Digits
BackToSchool
25.10.2022 00:37
Not that both $a$ and $b$ are even digits, $a \ge c$, and $a + b + c$ is multiple of $9$. Let $200 < \overline{abc} = 36 n< 900$ and we have $6 < n < 25$ Thus, we have $$\overline{abc} \in \{864, 612\}$$
f6700417
25.10.2022 09:10
We have $ 9 | a+b+c$, $ 4 | 10b+c $, $ 4 | 10b+a $, and $ a>c>0 $ . So $a, c$ are even, $4|a-c$. Thus, we have $ \overline{abc} = 864, 612 $.
f6700417
25.10.2022 09:11
BackToSchool wrote: Not that both $a$ and $b$ are even digits, $a \ge c$, and $a + b + c$ is multiple of $9$. Let $200 < \overline{abc} = 36 n< 900$ and we have $6 < n < 25$ Thus, we have $$\overline{abc} \in \{864, 828, 612, 252 \}$$ It seems that a≠c.