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2023 AMC 10B Problem 18

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2023 AMC 10B Problem 18AMC 10 Step by Step
Name:                             Date: September 13, 2026
GCD & LCM · difficulty 4 of 5 · about 5 min · position P16-20

Suppose aa , bb , and cc are positive integers such that a14+b15=c210.\dfrac{a}{14}+\dfrac{b}{15}=\dfrac{c}{210}.

Which of the following statements are necessarily true?

I. If gcd⁡(a,14)=1\gcd(a,14)=1 or gcd⁡(b,15)=1\gcd(b,15)=1 or both, then gcd⁡(c,210)=1\gcd(c,210)=1 .

II. If gcd⁡(c,210)=1\gcd(c,210)=1 , then gcd⁡(a,14)=1\gcd(a,14)=1 or gcd⁡(b,15)=1\gcd(b,15)=1 or both.

III. gcd⁡(c,210)=1\gcd(c,210)=1 if and only if gcd⁡(a,14)=gcd⁡(b,15)=1\gcd(a,14)=\gcd(b,15)=1 .

  1. A)

    I, II, and III\text{I, II, and III}

  2. B)

    I only\text{I only}

  3. C)

    I and II only\text{I and II only}

  4. D)

    III only\text{III only}

  5. E)

    II and III only\text{II and III only}

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