Difference between revisions of "2004 AMC 10B Problems/Problem 21"
(New page: ==Problem== Let <math>1</math>; <math>4</math>; <math>\ldots</math> and <math>9</math>; <math>16</math>; <math>\ldots</math> be two arithmetic progressions. The set <math>S</math> is the ...) |
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<math> \mathrm{(A) \ } 3722 \qquad \mathrm{(B) \ } 3732 \qquad \mathrm{(C) \ } 3914 \qquad \mathrm{(D) \ } 3924 \qquad \mathrm{(E) \ } 4007 </math> | <math> \mathrm{(A) \ } 3722 \qquad \mathrm{(B) \ } 3732 \qquad \mathrm{(C) \ } 3914 \qquad \mathrm{(D) \ } 3924 \qquad \mathrm{(E) \ } 4007 </math> | ||
| − | |||
| + | ==Solution 1== | ||
The two sets of terms are <math>A=\{ 3k+1 : 0\leq k < 2004 \}</math> and <math>B=\{ 7l+9 : 0\leq l<2004\}</math>. | The two sets of terms are <math>A=\{ 3k+1 : 0\leq k < 2004 \}</math> and <math>B=\{ 7l+9 : 0\leq l<2004\}</math>. | ||
| Line 18: | Line 18: | ||
Thus the good values of <math>l</math> are <math>\{1,4,7,\dots,856\}</math>, and their count is <math>858/3 = 286</math>. | Thus the good values of <math>l</math> are <math>\{1,4,7,\dots,856\}</math>, and their count is <math>858/3 = 286</math>. | ||
| − | Therefore <math>|A\cap B|=286</math>, and thus <math>|S|=4008-|A\cap B|=\boxed{3722}</math>. | + | Therefore <math>|A\cap B|=286</math>, and thus <math>|S|=4008-|A\cap B|=\boxed{(A) 3722}</math>. |
| + | ==Solution 2== | ||
| + | We can start by finding the first non-distinct term from both sequences. We find that that number is <math>16</math>. Now, to find every | ||
| + | |||
| + | other non-distinct terms, we can just keep adding <math>21</math>. We know that the last terms of both sequences are <math>1+3\cdot 2003</math> and | ||
| + | |||
| + | <math>9+7\cdot 2003</math>. Clearly, <math>1+3\cdot 2003</math> is smaller and that is the last possible common term of both sequences. Now, we can | ||
| + | |||
| + | create the inequality <math>16+21k \leq 1+3\cdot 2003</math>. Using the inequality, we find that there are <math>286</math> common terms. There are 4008 | ||
| + | |||
| + | terms in total. <math>4008-286=\boxed{(A) 3722}</math> | ||
| + | |||
| + | ~Hithere22702 | ||
== See also == | == See also == | ||
{{AMC10 box|year=2004|ab=B|num-b=20|num-a=22}} | {{AMC10 box|year=2004|ab=B|num-b=20|num-a=22}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 23:37, 24 May 2021
Contents
Problem
Let 
; 
; 
and 
; 
; be two arithmetic progressions. The set 
is the union of the first 
terms of each sequence. How many distinct numbers are in ?
Solution 1
The two sets of terms are 
and 
.
Now 
. We can compute 
. We will now find 
.
Consider the numbers in 
. We want to find out how many of them lie in 
. In other words, we need to find out the number of valid values of 
for which 
.
The fact "" can be rewritten as "
, and 
".
The first condition gives 
, the second one gives 
.
Thus the good values of are 
, and their count is 
.
Therefore 
, and thus 
.
Solution 2
We can start by finding the first non-distinct term from both sequences. We find that that number is . Now, to find every
other non-distinct terms, we can just keep adding 
. We know that the last terms of both sequences are 
and
. Clearly, is smaller and that is the last possible common term of both sequences. Now, we can
create the inequality 
. Using the inequality, we find that there are 
common terms. There are 4008
terms in total. 
~Hithere22702
See also
| 2004 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 20 |
Followed by Problem 22 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
