# Reversing Digits

Let $$N_{1}=\overline{abcd}$$ and $$N_{2}=\overline{dcba}$$ be two $$4$$ digits positive integers.

Let $$e$$ be a positive integer such that $$N_{1} \times e$$ = $$N_{2}$$.

How many ordered sets of integers $$a,b,c,d$$ and $$e$$ are there?

It's not necessary that $$a,b,c,d$$ are distinct.

$$A$$ $$request$$ : Please explain your steps a bit if you use Modular Arithmetic because I ain't much used to it.

Note by Abhimanyu Swami
4 years, 8 months ago

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Case-by-case analysis is not too bad. Consider $$e=1$$. Then from the original statement we know that $$a=d, b=c$$, which gives us $$9\cdot 10=90$$ sets.

Suppose now $$e>1$$. For the rest of the solution we have $$(a+1)e>d>a$$.

We then check for sets $$(a,d,e)$$ that satisfy the above inequality and $$a$$ is the last digit of $$de$$. Note that we just need to check for $$a<\frac{10}{e}$$, which is not a lot, that is if we fix $$a$$ and $$e$$, there are only $$12$$ valid pairs of $$(a, e)$$ just by $$1<a<\frac{10}{e}$$. One gets $$(a,d,e)=(2,8,4), (1,9,9)$$.

Case 1: $$(a,d,e)=(2,8,4)$$

From the original statement, we arrive at $$400b+40c+30=100c+10b$$ which simplifies to $$390b+30=60c$$. Therefore $$b<2$$. Case-by-case analysis for $$b=(0,1)$$ gives us only $$(b,c)=(1,7)$$, which gives us the ordered set $$(a,b,c,d,e)=(2,1,7,8,4)$$.

Case 2: $$(a,d,e)=(1,9,9)$$

Similarly, we have $$900b+90c+80=100c+10b$$ which simplifies to $$890b+80=10c$$. Therefore $$b=0$$, which gives $$c=8$$, which gives us the ordered set $$(a,b,c,d,e)=(1,0,8,9,9)$$.

We then conclude there are 92 such sets.

- 4 years, 8 months ago

Can you explain, why is $$(a+1)e>d$$, By the way, nice solution. Now can you please solve this one?

- 4 years, 8 months ago

Suppose $$d>ae+e$$. Then $$1000e\leq e\cdot \overline{bcd}\leq 999e$$, which is a contradiction.

- 4 years, 8 months ago

You can prove it by setting e equal to 1 or e larger than 1 so there are less steps involved

- 4 years, 8 months ago

Since $1234*9$ is more than $10000$ we find that e is less than 9. A case by case analysis might work here.

- 4 years, 8 months ago

But won't that be too long? And remember I didn't say that $$a,b,c,d$$ are distinct. They may or may not be.

- 4 years, 8 months ago