The seven-digit integer is a multiple of (where are digits). Determine all possible ordered pairs .

Source: NCTM Mathematics Teacher, December 2005

**SOLUTION**

Since and , we multiply by , etc. and by , etc. to find the possible digits that would match

Only and are close match of .

Verification

We do the same multiplication of by , etc.

Only is a close match

Verification

The possible ordered pairs are .

**Answer**:

**Alternative solution**

Since , must be divisible by and . To be divisible by the last three digits must be divisible by . To be divisible by the sum of the digits must be divisible by .

and are divisible by

Case 1:

not divisible by

No

Yes

No

Case 2:

not divisible by 8

No

Yes

No

No

No

Yes