For each of the following congruences, find all integers N, with N &gt; 1, that make the congruence true. a . 23 ≡ 13 ( mod N ) . b . 10 ≡ 5 ( mod N ) . c . 6 ≡ 60 ( mod N ) . d . 23 ≡ 22 ( mod N ) .

Question

For each of the following congruences, find all integers N, with N &amp;gt; 1, that make the congruence true.   a  .  23  ≡  13  (  mod  N  )  .  b  .  10  ≡  5  (  mod  N  )  .  c  .  6  ≡  60  (  mod  N  )  .  d  .  23  ≡  22  (  mod  N  )  .

Szulikto · Accepted Answer

a.   23  ≡  13  mod  (  N  ) if and only if N divides 23-13=10. So all the possible integers N&amp;gt;1 are 2,5,10b.   10  ≡  5  mod  (  N  ) if and only if N divides 10-5=5. So the only possible integer N&amp;gt;1 is 5c.   6  ≡  60  mod  (  N  ) if and only if N divides 6-60=54. So all the possible integers N&amp;gt;1 are 2,3,6,9,18,27,54d.   23  ≡  22  mod  (  N  ) if and only if N divides 23-22=1. There is no positive integer greater than 1 dividing 1. Therefore no such integer exists.Result:a. 2,5,10b. 5c. 2,3,6,9,18,27,54d. None

For each of the following congruences, find all integers N, with N > 1, that make the congruence true. a.23 equiv 13(mod N).b. 10equiv (mod N).c. 6equiv 60(mod N).d.23 equiv 22(mod N).

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