Algebraic Extensions of GCD and LCM

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| Questions: 20 | Updated: Nov 7, 2025
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1) GCD(3x, 12) = 3. Which x works?

Explanation

To keep only 3 as the GCD, 3x must be odd multiple of 3 (so x odd). From choices, x = 5 works (15 & 12 share only 3).

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About This Quiz
Algebraic Extensions Of GCD And LCM - Quiz

Ready to take GCD and LCM into algebra? In this quiz, you’ll solve equations that connect GCD and LCM to variables, test which values work, and discover how number properties extend into algebraic contexts. We bring you this quiz to help you see how arithmetic and algebra meet when working... see morewith divisibility, factors, and multiples.
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2) If x = 12, what is GCD(x, 18)?

Explanation

GCD(12, 18) = 6.

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3) Solve for x: LCM(4, x) = 28. Which x is correct?

Explanation

4 = 2², 28 = 2² × 7, so x must supply the 7 → x = 7.

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4) If GCD(20, x) = 10, which x values are possible?

Explanation

x must be a multiple of 10 but not of 20 (or else GCD would be 20). x = 30 works.

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5) If LCM(9, x) = 36, which x values are possible?

Explanation

9 = 3²; 36 = 2² × 3², so x needs a 2² factor → x = 12.

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6) Solve: GCD(18, x) = 9. Which x works?

Explanation

x must be a multiple of 9 but not share a factor 2 with 18 (otherwise GCD could be 18). x = 27 works.

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7) Solve: LCM(5, x) = 20. Which x is correct?

Explanation

5 and 20 give LCM 20 (note: 4 also works in general, but among choices, 20 is the one that works).

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8) Solve: GCD(8, x) = 4. Which x works?

Explanation

x must be a multiple of 4 but not of 8. x = 12 fits (gcd(8,12)=4).

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9) Which Property Is True?

Explanation

For two positive integers a, b: LCM(a, b) × GCD(a, b) = a × b.

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10) Solve linear equation: If LCM(3, x) = 21, then x = ?

Explanation

21 = 3 × 7, so x must be 7.

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11) If GCD(7, x) = 7, then x is always:

Explanation

That means x shares exactly a factor of 7 with 7, so x is a multiple of 7.

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12) If GCD(2x, 6) = 2, what is the smallest integer value of x?

Explanation

We need 2 as the only common factor with 6 (=2×3), so x must not be a multiple of 3. The smallest positive integer is x = 1.

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13) Solve for x: LCM(3, x) = 12. Which x works?

Explanation

We want the smallest x so that 3 and x reach 12. 3 and 4 give LCM 12.

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14) If LCM(2, x) = 14, then x = ?

Explanation

2 and 7 combine to 14 without extra primes.

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15) GCD(5x, 20) = 5. Which x values are possible?

Explanation

20 has factors 2² × 5. To keep the GCD at 5 (not 10), x must be odd and not introduce factor 2. From choices, x = 3 works.

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16) Solve: If GCD(12, x) = 6, which x values work?

Explanation

x must share exactly 6 as greatest common factor. x = 18 works (12 & 18 share 6; larger common factors don’t appear).

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17) If LCM(8, x) = 40, then x = ?

Explanation

8 = 2³; 40 = 2³ × 5, so x must bring in a 5 but not extra primes → x = 10.

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18) If GCD(14, x) = 7, then x could be:

Explanation

14 = 2 × 7. To have GCD 7, x must be a multiple of 7 but not even (otherwise GCD could be 14). x = 21 fits.

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19) Which is true?

Explanation

Both are multiples of 2a, so the greatest common divisor is 2a.

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20) LCM(6, x) = 18. Which x works?

Explanation

6 = 2 × 3; 18 = 2 × 3². We need an extra 3 → x = 9.

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GCD(3x, 12) = 3. Which x works?
If x = 12, what is GCD(x, 18)?
Solve for x: LCM(4, x) = 28. Which x is correct?
If GCD(20, x) = 10, which x values are possible?
If LCM(9, x) = 36, which x values are possible?
Solve: GCD(18, x) = 9. Which x works?
Solve: LCM(5, x) = 20. Which x is correct?
Solve: GCD(8, x) = 4. Which x works?
Which Property Is True?
Solve linear equation: If LCM(3, x) = 21, then x = ?
If GCD(7, x) = 7, then x is always:
If GCD(2x, 6) = 2, what is the smallest integer value of x?
Solve for x: LCM(3, x) = 12. Which x works?
If LCM(2, x) = 14, then x = ?
GCD(5x, 20) = 5. Which x values are possible?
Solve: If GCD(12, x) = 6, which x values work?
If LCM(8, x) = 40, then x = ?
If GCD(14, x) = 7, then x could be:
Which is true?
LCM(6, x) = 18. Which x works?
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