# 3 digit no.

Is there any 3 digit no. with distinct digits which when reversed divides the original no. perfectly? Can there be such a number for any n digits?

Note by Chandreyee Mitra
4 years, 8 months ago

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The only numbers that work are 510, 540, 810.

- 4 years, 8 months ago

According to my calculations there are no such 3-digit numbers.

For this problem we need to find $$a$$, $$b$$ and $$c$$, so that $$n*(100a+10b+c)=100*c+10b+a$$.

To start off we look at the last and first digits:

$$n*c \equiv a$$ (mod $$10$$) and $$n*a \lt 10$$. We make a multiplication table and cross out all impossible combinations.

We are left with;

$$n=1$$

Here $$a=c$$, so we don't have distinct digits.

$$n=2$$

Here $$a=2, c=6$$ or $$a=4, c=7$$.

$$n=3$$

Here $$a=2, c=4$$ or $$a=1, c=7$$

$$n=4$$

Here $$a=2, c=3$$ or $$a=2, c=8$$

$$n=7$$

Here $$a=1,c=3$$

$$n=9$$

Here $$a=1, c=9$$.

Substituting these 8 possibilities in the equation from the second line gives us 8 equations in $$b$$. None of these equations has a solution where $$0 \le b \le 9$$.

- 4 years, 8 months ago