Difference between revisions of "2022 AMC 12B Problems/Problem 3"
Bxiao31415 (talk | contribs) (Created page with "== Problem == How many of the first ten numbers of the sequence <math>121</math>, <math>11211</math>, <math>1112111</math>, ... are prime numbers? <math>\text{(A) } 0 \qquad \...") |
Bxiao31415 (talk | contribs) (→Problem) |
||
| Line 2: | Line 2: | ||
How many of the first ten numbers of the sequence <math>121</math>, <math>11211</math>, <math>1112111</math>, ... are prime numbers? | How many of the first ten numbers of the sequence <math>121</math>, <math>11211</math>, <math>1112111</math>, ... are prime numbers? | ||
<math>\text{(A) } 0 \qquad \text{(B) }1 \qquad \text{(C) }2 \qquad \text{(D) }3 \qquad \text{(E) }4</math> | <math>\text{(A) } 0 \qquad \text{(B) }1 \qquad \text{(C) }2 \qquad \text{(D) }3 \qquad \text{(E) }4</math> | ||
| + | |||
| + | == Solution 1 == | ||
| + | Let <math>P(a,b)</math> denote the digit <math>a</math> written <math>b</math> times and let <math>\overline{a_1a_2\cdots a_n}</math> denote the concatenation of <math>a_1</math>, <math>a_2</math>, ..., <math>a_n</math>. | ||
| + | Observe that <cmath>\overline{P(1,n) 2 P(1,n)} = \overline{P(1,n+1)P(0,n)} + P(1,n+1) = P(1,n+1) \cdot 10^n + P(1,n+1) = (P(1,n+1))(10^n + 1).</cmath> | ||
| + | Clearly, both terms are larger than <math>1</math> since <math>n \geq 1</math>, hence all the numbers of the sequence are <math>\fbox{0(A)}</math>, and we're done! | ||
Revision as of 18:15, 17 November 2022
Problem
How many of the first ten numbers of the sequence 
, 
, 
, ... are prime numbers?
Solution 1
Let 
denote the digit 
written 
times and let 
denote the concatenation of 
, 
, ..., 
.
Observe that ![\[\overline{P(1,n) 2 P(1,n)} = \overline{P(1,n+1)P(0,n)} + P(1,n+1) = P(1,n+1) \cdot 10^n + P(1,n+1) = (P(1,n+1))(10^n + 1).\]](http://latex.artofproblemsolving.com/a/9/8/a98263e7ccde384634ef4b630aa3715a5665cf56.png)
Clearly, both terms are larger than 
since 
, hence all the numbers of the sequence are 
, and we're done!
