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How many of the 1000 smallest positive integers are congruent to 1 modulo 9? Can you also explain what a modulo is?

 Jul 1, 2024
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Modulo is essentially the remainder we get when we divide two numbers.

For example, we have

\(10 \equiv 1 (\mod 3)\)

 

In this case, 1 is the remainder, 3 is the number we divide into 10. 

Thus, we say that 10 is congruent to 1 modulo 3. 

 

To solve the problem, we have the equation

\(9n+1 \equiv 1 (\mod 9)\)

 

We simply can write the equation

\(9n+1 \leq 1000\)

 

Now, we simplfy solve for n. Setting the two equations togther, we have

\(9n = 999\\ n=111\)

 

Thus, our answer is 111. 

 

Thanks! :)

 Jul 1, 2024

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