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# Number Theory

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How many of the 1000 smallest positive integers are congruent to 7 modulo 29?

Jul 2, 2024

#1
+1252
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We can write a handy equation to solve this problem.

First, let's set a variable. Let's set x as a positive integer.

Any number congruent to 7 modulo 29 can be written in the form

$$29x+7$$ since it is 7 greater than a factor of 29.

thus, we can write the inequality

$$29x+7\leq 1000$$

Setting the two to equal and solving for x, we have

$$29x+7=1000\\ 29x=993\\ x \approx 34.24$$

Rounding down, we find x is 34.

Thanks! :)

Jul 2, 2024

#1
+1252
+1

We can write a handy equation to solve this problem.

First, let's set a variable. Let's set x as a positive integer.

Any number congruent to 7 modulo 29 can be written in the form

$$29x+7$$ since it is 7 greater than a factor of 29.

thus, we can write the inequality

$$29x+7\leq 1000$$

Setting the two to equal and solving for x, we have

$$29x+7=1000\\ 29x=993\\ x \approx 34.24$$

Rounding down, we find x is 34.