Question

Call a 3-digit number geometric if it has $3$ distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest geometric numbers.

Answer

Solution:

Write the digits as $a,$ $ar,$ $ar^2.$ For the largest geometric number, take $a = 9.$ An integer ratio at least $2$ would push the next digit past $9,$ and $r = 1$ repeats digits, so $r$ is a fraction whose denominator squares into $9:$ the choices $r = \frac{2}{3}$ and $r = \frac{1}{3}$ give $964$ and $931.$ The largest is $964.$

For the smallest, take hundreds digit $1.$ Then the tens digit $r$ must be an integer at least $2$ (the digits are distinct), and $r = 2$ gives $124,$ which beats $r = 3$ 's $139.$

The difference is $964 - 124 = 840.$

2009 AIME I Problem 1

Solution:

Problem 1 in Other Years