ABC Murders by Agatha Christie

If two natural numbers $a$ ; $b$ have greatest common divisor equal to 1, then a; b are said to be relatively prime. The numbers $7$ , $8$ and $9$ are pairwise relatively prime, i.e. any pair are relatively prime. So their lowest common multiple is simply the product of all three. Written mathematically: $lcm(7, 8, 9)$ = $7 \times 8 \times 9$ = $504$ We must choose a number of the form $739ABC$ such that it is a multiple of $7$ , $8$ and $9$ ; i.e. we must choose a number of the form $739ABC$ that is divisible by $lcm(7, 8, 9)$ = $504$ . Now $739 000$ gives remainder $136$ on division by $504$ . Hence the numbers $739ABC$ we are looking for, are of form $739 000 - 136 + k.504$ where $k$ is an integer. We can see that $k$ can only be $1$ or $2$ . If $k = 1$ , we get the number $739 368$ so that one solution for $A$ ; $B$ ; $C$ is $A = 3$ ; $B = 6$ ; $C = 8$ ; and if $k$ = $2$ we get the number $739 872$ so that another solution for $A$ ; $B$ ; $C$ is $A$ = $8$ ; $B$ = $7$ ; $C$ = $2$

$3+6+8+8+7+2$ = $34$

• lcm(7,8,9)=7 8 9=504
• This means that the numbers between 7400 and 739 000 that might be divisible by 504 are as maximum 2.
• The first number is 504 * 1467 = 739 368 (A=3, B=6, C=8)
• The second number is 504 * 1468 = 739 872 (A=8, B=7, C=2)
• 3+6+8+8+7+2=34