Difference between revisions of "2022 AMC 12B Problems/Problem 3"
Ehuang0531 (talk | contribs) m (→Solution 1) |
|||
Line 6: | Line 6: | ||
Write <math>121 = 110 + 11 = 11(10+1)</math>, <math>11211 = 11100 + 111 = 111(100+1)</math>, <math>1112111 = 1111000 + 1111 = 1111(1000+1)</math>. It becomes clear that <math>\boxed{\textbf{(A) } 0}</math> of these numbers are prime. | Write <math>121 = 110 + 11 = 11(10+1)</math>, <math>11211 = 11100 + 111 = 111(100+1)</math>, <math>1112111 = 1111000 + 1111 = 1111(1000+1)</math>. It becomes clear that <math>\boxed{\textbf{(A) } 0}</math> of these numbers are prime. | ||
− | + | In general, <math>11...121...1</math> (where there are <math>k</math> <math>1</math>'s on either side of the <math>2</math>) can be written as <math>(11...11)10^k + 11...11 = 11...11(10^k + 1)</math>, where the first term has <math>(k + 1)</math> <math>1</math>'s. | |
== Solution 2 == | == Solution 2 == |
Revision as of 22:16, 17 November 2022
Contents
Problem
How many of the first ten numbers of the sequence , , , ... are prime numbers?
Solution 1
Write , , . It becomes clear that of these numbers are prime.
In general, (where there are 's on either side of the ) can be written as , where the first term has 's.
Solution 2
Let denote the digit written times and let denote the concatenation of , , ..., .
Observe that
Both terms are integers larger than since , so of the numbers of the sequence are prime.
See Also
2022 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 2 |
Followed by Problem 4 |
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 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.