Divide and conquer

Number Theory Level 4

Let $\overline{ABC}$ be a three digit positive integer with $A>C \ge 0$ and having the property that

$\overline{ABC} \equiv 0 \mod \overline{CBA}$ .

If $n_1,n_2,\cdots,n_k$ are $k$ numbers satisfying the above constraints,

Find $n_1+n_2+\cdots+n_k$

1 solution

Janardhanan Sivaramakrishnan
Aug 19, 2015

All numbers of the form $\overline{A00}$ are divisible by $\overline{00A}=A$ .

This gives 9 solutions $100,200,\cdots,900$

Further, all numbers of form $\overline{AA0}$ are divisible by $\overline{0AA}=\overline{AA}$

This gives a further 9 solutions $110,220,\cdots,990$

In addition to these, we have three more solutions

$510 \equiv 0 \mod 15$

$540 \equiv 0 \mod 45$

$810 \equiv 0 \mod 18$

Adding all the 21 solutions gives the required answer as $\boxed{11310}$