Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2008 iTest Problems/Problem 39"

(→‎Proof)
(→‎Problem)
 
(7 intermediate revisions by the same user not shown)
Line 1: Line 1:
 
==Problem==
 
==Problem==
  
−
Let <math>\phi(n)</math> denote <math>\textit{Euler's phi function}</math>, the number of integers <math>1\leq i\leq n</math> that are relatively prime to <math>n</math>. (For example, <math>\phi(6)=2</math> and <math>\phi(10)=4</math>.)  
+
Let <math>\phi(n)</math> denote <math>\textit{Euler's phi function}</math>, the number of integers <math>1\leq i\leq n</math> that are relatively prime with <math>n</math>. (For example, <math>\phi(6)=2</math> and <math>\phi(10)=4</math>.)  
−
Let <math>S=\sum_{d|2008}\phi(d)</math>, in which <math>d</math> ranges through all positive divisors of <math>2008</math>, including <math>1</math> and <math>2008</math>. Find the remainder when <math>S</math> is divided by <math>1000</math>.  
+
Let <math>S=\sum_{d|2008}\phi(d)</math>, where <math>d</math> ranges through all positive divisors of <math>2008</math>, including <math>1</math> and <math>2008</math>. Find the remainder when <math>S</math> is divided by <math>1000</math>.
  
 
==Solution==
 
==Solution==
Line 12: Line 12:
 
==Extra Note==
 
==Extra Note==
 
If all the divisors of an integer n are {<math>{d_1}</math>,  <math>{d_2}</math>,  <math>{d_3}</math>, ... <math>{d_k}</math>}, then n  =  <math>\phi({d_1})</math> +  <math>\phi({d_2})</math> +  .... +  <math>\phi({d_k})</math>.
 
If all the divisors of an integer n are {<math>{d_1}</math>,  <math>{d_2}</math>,  <math>{d_3}</math>, ... <math>{d_k}</math>}, then n  =  <math>\phi({d_1})</math> +  <math>\phi({d_2})</math> +  .... +  <math>\phi({d_k})</math>.
−
This provides a shorter approach in this problem because it asks about the sum of the number of numbers co prime to 2008's divisors, as mentioned earlier. This is the same as 2008 itself, drastically simplifying the problem.
+
This provides a shorter approach in this problem because it asks about the sum of the number of numbers co prime to 2008's divisors, as mentioned earlier. This is 2008 itself.
  
 
==Proof==
 
==Proof==
  
−
Given that: <math>n</math> = <math>{p_1}^{\alpha_1}</math><math>{p_2}^{\alpha_2}</math>....<math>{p_k}^{\alpha_k}</math>
+
Let <math>n</math> = <math>{p_1}^{\alpha_1}</math><math>{p_2}^{\alpha_2}</math>....<math>{p_k}^{\alpha_k}</math> where <math>{p_1}, {p_2}, ... {p_k}</math> are prime numbers
  
 
and  
 
and  
  
−
<math>{d_1}</math>, <math>{d_2}</math>, .... <math>{d_m}</math> are all of <math>n</math>'s divisors
+
{<math>{d_1}</math>, <math>{d_2}</math>, .... <math>{d_m}</math>} is an exhaustive, mutually exclusive list of <math>n</math>'s factors
  
 
<math>\phi({d_1})</math> + <math>\phi({d_2})</math> + ... + <math>\phi({d_m})</math>  
 
<math>\phi({d_1})</math> + <math>\phi({d_2})</math> + ... + <math>\phi({d_m})</math>  
Line 32: Line 32:
 
= <math>(1 + (1 - \frac{1}{p_1})</math><math>({p_1} + {p_1}^{2} + ... + {p_1}^{\alpha_1})</math>)<math>(1 + (1 - \frac{1}{p_2})</math><math>({p_2} + {p_2}^{2} + ... + {p_2}^{\alpha_2})</math>) ... <math>(1 + (1 - \frac{1}{p_k})</math><math>({p_k} + {p_k}^{2} + ... + {p_k}^{\alpha_k})</math>
 
= <math>(1 + (1 - \frac{1}{p_1})</math><math>({p_1} + {p_1}^{2} + ... + {p_1}^{\alpha_1})</math>)<math>(1 + (1 - \frac{1}{p_2})</math><math>({p_2} + {p_2}^{2} + ... + {p_2}^{\alpha_2})</math>) ... <math>(1 + (1 - \frac{1}{p_k})</math><math>({p_k} + {p_k}^{2} + ... + {p_k}^{\alpha_k})</math>
  
−
= <math>(1 + {p_1} + {p_1}^{2} + ... + {p_1}^{\alpha_1} - (1 + {p_1} + {p_1}^{2} + ... + {p_1}^{\alpha_1 - 1}))</math>
+
= <math>(1 + {p_1} + {p_1}^{2} + ... + {p_1}^{\alpha_1} - (1 + {p_1} + {p_1}^{2} + ... + {p_1}^{\alpha_1 - 1}))</math> x
  
−
<math>(1 + {p_2} + {p_2}^{2} + ... + {p_2}^{\alpha_2} - (1 + {p_2} + {p_2}^{2} + ... + {p_2}^{\alpha_2 - 1}))</math>...
+
<math>(1 + {p_2} + {p_2}^{2} + ... + {p_2}^{\alpha_2} - (1 + {p_2} + {p_2}^{2} + ... + {p_2}^{\alpha_2 - 1}))</math> x ...
  
 
<math>(1 + {p_k} + {p_k}^{2} + ... + {p_k}^{\alpha_k} - (1 + {p_k} + {p_k}^{2} + ... + {p_k}^{\alpha_k - 1}))</math>
 
<math>(1 + {p_k} + {p_k}^{2} + ... + {p_k}^{\alpha_k} - (1 + {p_k} + {p_k}^{2} + ... + {p_k}^{\alpha_k - 1}))</math>

Latest revision as of 19:49, 22 August 2023

Problem

Let $\phi(n)$ denote $\textit{Euler's phi function}$, the number of integers $1\leq i\leq n$ that are relatively prime with $n$. (For example, $\phi(6)=2$ and $\phi(10)=4$.) Let $S=\sum_{d|2008}\phi(d)$, where $d$ ranges through all positive divisors of $2008$, including $1$ and $2008$. Find the remainder when $S$ is divided by $1000$.

Solution

The divisors of $2008$ are $1$,$2$,$4$,$8$,$251$,$502$,$1004$, and $2008$, so the problem requires the sum of the number of numbers relatively prime to each of the eight numbers.

Using the formula $\phi(n) = n \cdot (1 - \tfrac{1}{p_1})(1 - \tfrac{1}{p_2}) \cdots (1 - \tfrac{1}{p_n})$ (or by manually counting the numbers relatively prime to each number), $1$ number is relatively prime to $1$, $1$ number is relatively prime to $2$, $2$ numbers are relatively prime to $4$, $4$ numbers are relatively prime to $8$, $250$ numbers are relatively prime to $251$, $250$ numbers are relatively prime to $502$, $500$ numbers are relatively prime to $1004$, and $1000$ numbers are relatively prime to $2008$. That means $S = 2008$, and the remainder when $S$ is divided by $1000$ is $\boxed{8}$.

Extra Note

If all the divisors of an integer n are {${d_1}$, ${d_2}$, ${d_3}$, ... ${d_k}$}, then n = $\phi({d_1})$ + $\phi({d_2})$ + .... + $\phi({d_k})$. This provides a shorter approach in this problem because it asks about the sum of the number of numbers co prime to 2008's divisors, as mentioned earlier. This is 2008 itself.

Proof

Let $n$ = ${p_1}^{\alpha_1}$${p_2}^{\alpha_2}$....${p_k}^{\alpha_k}$ where ${p_1}, {p_2}, ... {p_k}$ are prime numbers

and

{${d_1}$, ${d_2}$, .... ${d_m}$} is an exhaustive, mutually exclusive list of $n$'s factors

$\phi({d_1})$ + $\phi({d_2})$ + ... + $\phi({d_m})$

= ${p_1}(1-\frac{1}{p_1})$ + ${p_2}(1-\frac{1}{p_2})$ + ${p_1} {p_2}(1-\frac{1}{p_1})(1-\frac{1}{p_2})$...

The above term represents the sum of all the values of the the totient function applied of all of n's divisors, like the problem is asking.

= $(1 - \frac{1}{p_1})$$(1 - \frac{1}{p_2})$ $({p_1} + {p_1}^{2} + ... + {p_1}^{\alpha_1})$ $({p_2} + {p_2}^{2} + ... + {p_2}^{\alpha_2})$ + ....

= $(1 + (1 - \frac{1}{p_1})$$({p_1} + {p_1}^{2} + ... + {p_1}^{\alpha_1})$)$(1 + (1 - \frac{1}{p_2})$$({p_2} + {p_2}^{2} + ... + {p_2}^{\alpha_2})$) ... $(1 + (1 - \frac{1}{p_k})$$({p_k} + {p_k}^{2} + ... + {p_k}^{\alpha_k})$

= $(1 + {p_1} + {p_1}^{2} + ... + {p_1}^{\alpha_1} - (1 + {p_1} + {p_1}^{2} + ... + {p_1}^{\alpha_1 - 1}))$ x

$(1 + {p_2} + {p_2}^{2} + ... + {p_2}^{\alpha_2} - (1 + {p_2} + {p_2}^{2} + ... + {p_2}^{\alpha_2 - 1}))$ x ...

$(1 + {p_k} + {p_k}^{2} + ... + {p_k}^{\alpha_k} - (1 + {p_k} + {p_k}^{2} + ... + {p_k}^{\alpha_k - 1}))$

= ${p_1}^{\alpha_1}$${p_2}^{\alpha_2}$ ... ${p_k}^{\alpha_k}$

= $n$

See Also

2008 iTest (Problems)
Preceded by:
Problem 38 		Followed by:
Problem 40
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 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60 • 61 • 62 • 63 • 64 • 65 • 66 • 67 • 68 • 69 • 70 • 71 • 72 • 73 • 74 • 75 • 76 • 77 • 78 • 79 • 80 • 81 • 82 • 83 • 84 • 85 • 86 • 87 • 88 • 89 • 90 • 91 • 92 • 93 • 94 • 95 • 96 • 97 • 98 • 99 • 100
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2008_iTest_Problems/Problem_39&oldid=197313"