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741
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avatar+109 

(a) How many positive integers \(N\)  from 1 to 5000 satisfy the congruence \(N\equiv5\pmod{12}\) ?

 

(b) How many positive integers \(N\)  from 1 to 5000 satisfy the congruence \(N\equiv11\pmod{13}\) ?

 Sep 14, 2017
edited by Jeff123  Sep 14, 2017
 #1
avatar+22577 
+5

Modular Arithmetic

 

(a) How many positive integers   from 1 to 5000 satisfy the congruence  \(N\equiv5\pmod{12} \)?

\(\begin{array}{|lrcll|} \hline N\equiv5\pmod{12} & \text { or } & N-5 &=& n\cdot 12 \\ & & N &=& n\cdot 12 + 5 \quad & | \quad N_{max} = 5000 \\ & & 5000 &=& n\cdot 12 + 5 \\ & & n &=& \frac{5000-5}{12} \\ & & n &=& [416].25 \\\\ \mathbf{n=416} &\Rightarrow& N &=& 416\cdot 12 + 5 = 4997 \\ \hline \end{array}\)

 

416  + 1(n=0)  = 417 positive integers from 1 to 5000 satisfy the congruence \(N\equiv5\pmod{12}\)

 

 

(b) How many positive integers   from 1 to 5000 satisfy the congruence \( N\equiv11\pmod{13}\) ?

\(\begin{array}{|lrcll|} \hline N\equiv11\pmod{13} & \text { or } & N-11 &=& n\cdot 13 \\ & & N &=& n\cdot 13 + 11 \quad & | \quad N_{max} = 5000 \\ & & 5000 &=& n\cdot 13 + 11 \\ & & n &=& \frac{5000-11}{13} \\ & & n &=& [383].769230769 \\\\ \mathbf{n=383} &\Rightarrow & N &=& 383\cdot 13 + 11 = 4990 \\ \hline \end{array}\)

 

383 + 1(n=0) = 384 positive integers from 1 to 5000 satisfy the congruence \(N\equiv11\pmod{13}\)

 

laugh

 Sep 14, 2017
edited by heureka  Sep 15, 2017
 #2
avatar+109 
+1

The answer is wrong, I tried those answers before too. IDK why it won't work...

 Sep 14, 2017
 #3
avatar
+2

Both answers above forget to include n = 0.

Answers should be 417 and 384 resp.

 Sep 15, 2017
 #4
avatar+22577 
+4

Thank you!

 

blushlaugh

heureka  Sep 15, 2017
 #5
avatar+109 
+1

Thank you so much!!!!!!

 Sep 16, 2017

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