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# Modular Arithmetic Help!!!

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(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 $$N\equiv11\pmod{13}$$ ?

Jeff123  Sep 14, 2017
edited by Jeff123  Sep 14, 2017
Sort:

#1
+19095
+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 $$N\equiv11\pmod{13}$$

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

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

Jeff123  Sep 14, 2017
#3
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Both answers above forget to include n = 0.

Answers should be 417 and 384 resp.

Guest Sep 15, 2017
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+19095
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Thank you!

heureka  Sep 15, 2017
#5
+109
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Thank you so much!!!!!!

Jeff123  Sep 16, 2017

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