Difference between revisions of "2009 AIME I Problems/Problem 1"

Latest revision as of 02:32, 16 January 2023

Problem

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.

Solution 1

Assume that the largest geometric number starts with a $9$ . We know that the common ratio must be a rational of the form $k/3$ for some integer $k$ , because a whole number should be attained for the 3rd term as well. When $k = 1$ , the number is $931$ . When $k = 2$ , the number is $964$ . When $k = 3$ , we get $999$ , but the integers must be distinct. By the same logic, the smallest geometric number is $124$ . The largest geometric number is $964$ and the smallest is $124$ . Thus the difference is $964 - 124 = \boxed{840}$ .

Solution 2

Consider the three-digit number $abc$ . If its digits form a geometric progression, we must have that ${a \over b} = {b \over c}$ , that is, $b^2 = ac$ .

The minimum and maximum geometric numbers occur when $a$ is minimized and maximized, respectively. The minimum occurs when $a = 1$ ; letting $b = 2$ and $c = 4$ achieves this, so the smallest possible geometric number is 124.

For the maximum, we have that $b^2 = 9c$ ; $b$ is maximized when $9c$ is the greatest possible perfect square; this happens when $c = 4$ , yielding $b = 6$ . Thus, the largest possible geometric number is 964.

Our answer is thus $964 - 124 = \boxed{840}$ .

Solution 3

The smallest geometric number is $124$ because $123$ and any number containing a zero does not work. $964$ is the largest geometric number because the middle digit cannot be 8 or 7. Subtracting the numbers gives $\boxed{840}.$

Video Solution by OmegaLearn

https://youtu.be/1-iWPCWPsLw?t=195

~ pi_is_3.14

Video Solution

https://youtu.be/NL79UexadzE

~IceMatrix

Video Solution 2

https://www.youtube.com/watch?v=P00iOJdQiL4

~Shreyas S

@@ Line 1: / Line 1: @@
 == Problem ==
 Call a <math>3</math>-digit number ''geometric'' if it has <math>3</math> distinct digits which, when read from left to right, form a geometric sequence.  Find the difference between the largest and smallest geometric numbers.
-== Solution ==
+== Solution 1 ==
+Assume that the largest geometric number starts with a <math>9</math>. We know that the common ratio must be a rational of the form <math>k/3</math> for some integer <math>k</math>, because a whole number should be attained for the 3rd term as well. When <math>k = 1</math>, the number is <math>931</math>. When <math>k = 2</math>, the number is <math>964</math>. When <math>k = 3</math>, we get <math>999</math>, but the integers must be distinct. By the same logic, the smallest geometric number is <math>124</math>. The largest geometric number is <math>964</math> and the smallest is <math>124</math>. Thus the difference is <math>964 - 124 = \boxed{840}</math>.
+== Solution 2 ==
+Consider the three-digit number <math>abc</math>. If its digits form a geometric progression, we must have that <math>{a \over b} = {b \over c}</math>, that is, <math>b^2 = ac</math>.
+The minimum and maximum geometric numbers occur when <math>a</math> is minimized and maximized, respectively. The minimum occurs when <math>a = 1</math>; letting <math>b = 2</math> and <math>c = 4</math> achieves this, so the smallest possible geometric number is 124.
+For the maximum, we have that <math>b^2 = 9c</math>; <math>b</math> is maximized when <math>9c</math> is the greatest possible perfect square; this happens when <math>c = 4</math>, yielding <math>b = 6</math>. Thus, the largest possible geometric number is 964.
+Our answer is thus <math>964 - 124 = \boxed{840}</math>.
+==Solution 3==
+The smallest geometric number is <math>124</math> because <math>123</math> and any number containing a zero does not work. <math>964</math> is the largest geometric number because the middle digit cannot be 8 or 7. Subtracting the numbers gives <math>\boxed{840}.</math>
+== Video Solution by OmegaLearn ==
+https://youtu.be/1-iWPCWPsLw?t=195
+~ pi_is_3.14
+==Video Solution==
+https://youtu.be/NL79UexadzE
+~IceMatrix
+==Video Solution 2==
+https://www.youtube.com/watch?v=P00iOJdQiL4
-The largest geometric number is 964 and the smallest is 124. Thus the difference is 964 - 124 = 840.
+~Shreyas S
 == See also ==
@@ Line 12: / Line 39: @@
 [[Category:Intermediate Number Theory Problems]]
+{{MAA Notice}}

2009 AIME I (Problems • Answer Key • Resources)
Preceded by First Question	Followed by Problem 2
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

Page

Toolbox

Search