Difference between revisions of "2004 AMC 10B Problems/Problem 21"
Scrabbler94 (talk | contribs) (→Solution: solution 2 isn't really correct as the positions of the terms which appear in both sequences change after shifting.) |
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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>. |
== See also == | == See also == |
Revision as of 19:34, 26 January 2020
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
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 .
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 |
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