Difference between revisions of "2008 iTest Problems/Problem 48"
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We have <math>\underbrace{111 \cdots 1}_{n\ \text{ones}} = 10^{n-1} + 10^{n-2} + \cdots + 1 = \frac{10^n - 1}{9}</math>. Since <math>9</math> and <math>97</math> are [[relatively prime]], it suffices for <math>10^{n} - 1 \equiv 0 \pmod{97}</math>, or <math>10^{n} \equiv 1 \pmod{97}</math>. | We have <math>\underbrace{111 \cdots 1}_{n\ \text{ones}} = 10^{n-1} + 10^{n-2} + \cdots + 1 = \frac{10^n - 1}{9}</math>. Since <math>9</math> and <math>97</math> are [[relatively prime]], it suffices for <math>10^{n} - 1 \equiv 0 \pmod{97}</math>, or <math>10^{n} \equiv 1 \pmod{97}</math>. | ||
| − | The smallest integer <math>m</math> such that <math>a^m \equiv 1 \pmod{p}</math> where <math>\text{gcd}\,(a,p)=1</math> is <math>m = \lambda (p)</math>, where <math>\lambda (p)</math> is the [[Carmichael function]]. For primes <math>p</math>, <math>\lambda(p) = p-1</math>, and so the answer is <math>n = \boxed{96}</math>. | + | The smallest integer <math>m</math> such that <math>a^m \equiv 1 \pmod{p}</math> where <math>\text{gcd}\,(a,p)=1</math> is <math>m = \lambda (p)</math>, where <math>\lambda (p)</math> is the [[Carmichael function]]. For primes <math>p</math>, <math>\lambda(p) = p-1</math>, and so the answer is <math>n = \boxed{96}</math>. |
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| + | {{2008 iTest box|num-b=47|num-a=49}} | ||
== See also == | == See also == | ||
[[Category:Intermediate Number Theory Problems]] | [[Category:Intermediate Number Theory Problems]] | ||
Revision as of 18:33, 12 July 2018
Problem
A repunit is a natural number whose digits are all 
. For instance,
![\[1,11,111,1111, \ldots\]](http://latex.artofproblemsolving.com/6/3/8/6386858630e6455adad0971f9ba1465e78468343.png)
are the four smallest repunits. How many digits are there in the smallest repunit that is
divisible by 
?
Solution
We have 
. Since 
and are relatively prime, it suffices for 
, or 
.
The smallest integer 
such that 
where 
is 
, where 
is the Carmichael function. For primes 
, 
, and so the answer is 
.
| 2008 iTest (Problems) | ||
| Preceded by: Problem 47 |
Followed by: Problem 49 | |
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