B3. The letters $a, b$ and $c$ stand for non-zero digits. The integer ' $a b c$ ' is a multiple of 3 ; the integer ' $c b a b c$ ' is a multiple of 15 ; and the integer ' $a b c b a$ ' is a multiple of 8 .
What is the integer ' $a b c$ '?
Solution
The question tells us that ' $a b c b a$ ' is a multiple of 8 . Therefore, since 1000 and hence any multiple of 1000 , is a multiple of 8 , ' $c b a$ ' is a multiple of 8 .
The question also tells us that ' $a b c$ ' is a multiple of 3 and hence, since the digit sums of ' $a b c$ ' and ' $c b a$ ' are the same, ' $c b a$ ' is also a multiple of 3 . Hence, ' $c b a$ ' is a multiple of 24 .
Finally, we are told that ' $c b a b c$ ' is a multiple of 15 . Since $c \neq 0$, then $c=5$ and $c+b$ is a multiple of 3 .
The three-digit multiples of 24 which are the possible values of ' $c b a$ ' beginning with ' 5 ' are 504, 528,552 and 576 and, of these, only 576 has $c+b$ as a multiple of 3 .
Therefore the integer ' $a b c$ ' is 675 .