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Find all 4 digits numbers $\overline{abcd}$ that $\overline{abcd}=20\cdot\overline{ab}+16\cdot\overline{cd}$.

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    what does bar represent?? – 2017-01-27
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    In right hand $bc$ or $cd$.? – 2017-01-27
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    My guess is that it is the base-10 number obtained by concatenating digits - e.g. $9876 = 20 \cdot 96 + 16 \cdot 87$ (not actually equal BTW) – 2017-01-27
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    @user8795 for example $\overline{abcd}$ means $1000a+100b+10c+d$ – 2017-01-27
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    @Snip3r did you put any effort into this problem so far? – 2017-01-27
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    solve the equation $$d-6c-80b+800a=0$$ whre $$0$$0\le b,c,d\le 9$$ – 2017-01-27
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    I just now realized, that I have mistake in question. Question is correct now. Sorry for trouble. – 2017-01-27

2 Answers 2

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We have:

$$\overline{abcd} = 20 \cdot \overline{ab} + 16 \cdot \overline{bc}$$ $$1000a + 100b + 10c + d = 200a + 20b + 160b + 16c$$ $$800a + d = 80b + 6c$$

Note that $a\ge 1$, hence: $720 + 54 \ge 80b + 6c = 800a + d \ge 800$.

But this is impossible, so therefore no solutions.

Considering the edit we have:

$$\overline{abcd} = 20 \cdot \overline{ab} + 16 \cdot \overline{cd}$$ $$1000a + 100b + 10c + d = 200a + 20b + 160c + 16d$$ $$800a + 80b = 150c + 15d$$

We have $a=1$, as otherwise: $1350 + 135 \ge 150c + 15d = 800a + 8b \ge 1600$

Now divide by $5$ to get: $160 + 16b = 30c + 3d$. Now modulo $3$ we have that $b = 2,5,8$. All these produce a solution and $c,d$ are given by $\overline{cd} = \frac{16\cdot \overline{1b}}{3}$.

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Hint: let $m = \overline{ab}$ and $n = \overline{cd}$ then $100m+n=20m+16n \iff 80 m = 15n \iff 16m=3n$. Therefore $n$ must be a multiple of $16$ and since $n \le 99$ this leaves only a few values to try. Adding the condition $m \ge 10$ (so that $\overline{abcd}$ has in fact $4$ digits) gives the solutions $1264$, $1580$, $1896$.