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How many integers m are there such that 0 < m < 100 and gcd(m, 1001) is a single-digit number?

 Apr 15, 2022
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The Greatest Common Divisor has to be either 1 or 7. 

 

If the GCD is 7, there are 12 options form: \(7,14,21,28,35,42,49,56,63,70,84,98\) (Note: it can't be 77 or 91, because both are factors of 1001)

 

If the GCD is 1, m must be a number without the factors 7, 11, or 13.

 

There are \(100-(14+9+7-2) = 72\) numbers that meet these criteria

 

Thus, the answer is \(72+12=\color{brown}\boxed{84}\)

 Apr 15, 2022

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