Question

How many three-digit positive integers class="latex-formula"> satisfy $3874x+481\equiv 1205 \pmod{23}$

Answer 1

Answer:

  40

Step-by-step explanation:

Any solution x will mod 23 will also have x+23n as a solution, for some integer n. Since 900/23 = 39 3/23, we know there are 39 or 40 three-digit integers of this form.

As it happens, 100 is the smallest 3-digit solution. So, there are 40 three-digit numbers that are of the form 100 +23n, hence 40 solutions to the equation.

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The equation reduces, mod 23, to ...

  10x = 11

Its solutions are x = 23n +8.