Difference between revisions of "Mock AIME 6 2006-2007 Problems/Problem 1"
(Created page with "We note that the required sum is equal to <math>(2^0+2^1+2^2+2^3+2^4)(3^0+3^1+3^2+3^3+3^4)=(31)(121)=3751</math>, which is <math>\framebox{751}</math> mod 1000.") |
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| + | ==Problem== | ||
| + | |||
| + | Let <math>T</math> be the sum of all positive integers of the form <math>2^r\cdot3^s</math>, where <math>r</math> and <math>s</math> are nonnegative integers that do not exceed <math>4</math>. Find the remainder when <math>T</math> is divided by <math>1000</math>. | ||
| + | |||
| + | ==Solution== | ||
| + | |||
We note that the required sum is equal to <math>(2^0+2^1+2^2+2^3+2^4)(3^0+3^1+3^2+3^3+3^4)=(31)(121)=3751</math>, | We note that the required sum is equal to <math>(2^0+2^1+2^2+2^3+2^4)(3^0+3^1+3^2+3^3+3^4)=(31)(121)=3751</math>, | ||
which is <math>\framebox{751}</math> mod 1000. | which is <math>\framebox{751}</math> mod 1000. | ||
| + | ~AbbyWong | ||
Latest revision as of 09:17, 23 November 2023
Problem
Let 
be the sum of all positive integers of the form 
, where 
and 
are nonnegative integers that do not exceed 
. Find the remainder when is divided by 
.
Solution
We note that the required sum is equal to 
,
which is 
mod 1000.
~AbbyWong
