Competition problem — Number theory

Library / /3 of 15

Number theory Difficulty 4.5 AIME Prove it Philippines

Problem:

The 6-digit number $739ABC$ is divisible by $7$ , $8$ , and $9$ . What values can $A$ , $B$ , and $C$ take?

Solution

Solution:

Since $\gcd(7,8,9) = 1$ , then $739ABC$ is divisible by $7$ , $8$ , and $9$ if and only if it is divisible by $7 \cdot 8 \cdot 9 = 504$ . Note that the only integers between $739000$ and $739999$ which are divisible by $504$ are $739368$ and $739872$ . So, $(A, B, C) \in \{(3,6,8), (8,7,2)\}$ .

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.