Coprime Integers in Algebra: Linear Combinations and Properties

1) Solve for x: GCD(7, x) = 1. Which x works?

14

21

20

28

7 is prime; 20 = 2² × 5 → no shared factors → coprime.

Explanation

7 is prime; 20 = 2² × 5 → no shared factors → coprime.

2) If a and b are coprime, then ax + by = 1 has:

No solution

One solution

Integer solutions

Only rational solutions

That’s a key result in number theory (Bezout’s Identity).

Explanation

That’s a key result in number theory (Bezout’s Identity).

3) Solve: If GCD(15, x) = 1, which value of x works?

20

22

25

30

15 = 3 × 5; 22 = 2 × 11 → no overlap → coprime.

Explanation

15 = 3 × 5; 22 = 2 × 11 → no overlap → coprime.

4) Which pair is not coprime?

(7, 15)

(8, 21)

(9, 27)

(11, 20)

Both share 3’s as factors.

Explanation

Both share 3’s as factors.

5) Solve: If GCD(21, x) = 1, which value of x works?

14

28

35

20

21 = 3 × 7; 20 = 2² × 5 → no overlap → coprime.

Explanation

21 = 3 × 7; 20 = 2² × 5 → no overlap → coprime.

6) Which of these pairs are coprime?

(4, 9)

(8, 15)

(11, 28)

All of the above

(4,9), (8,15), and (11,28) have no shared factors.

Explanation

(4,9), (8,15), and (11,28) have no shared factors.

7) Solve: GCD(18, x) = 1. Which x works?

6

9

11

12

18 = 2 × 3²; 11 is prime and unrelated → coprime.

Explanation

18 = 2 × 3²; 11 is prime and unrelated → coprime.

8) If a and b are coprime, what is LCM(a, b)?

A + b

GCD(a, b)

A × b

A – b

They have no shared primes, so LCM equals their product.

Explanation

They have no shared primes, so LCM equals their product.

9) If a and b are coprime, then GCD(a², b²) = ?

A

B

Ab

1

Their squares don’t add shared factors.

Explanation

Their squares don’t add shared factors.

10) Which of these pairs is not coprime?

(5, 12)

(7, 20)

(9, 16)

(8, 12)

Both share factor 4 → not coprime.

Explanation

Both share factor 4 → not coprime.

11) Which statement is true?

All prime pairs are coprime

All distinct primes are coprime

Coprime numbers must both be prime

Coprime numbers must be odd

All distinct primes are coprime.

Explanation

All distinct primes are coprime.

12) Which pair is coprime?

(9, 12)

(12, 15)

(16, 24)

(13, 20)

13 = prime; 20 = 2² × 5 → no shared factors → coprime.

Explanation

13 = prime; 20 = 2² × 5 → no shared factors → coprime.

13) If GCD(a, b) = 1, which of these equations has integer solutions?

Ax – b = 0

A + b = 1

Ax + by = 1

Ab = 1

If GCD(a, b) = 1 → ax + by = 1 has integer solutions.

Explanation

If GCD(a, b) = 1 → ax + by = 1 has integer solutions.

14) Which pair is coprime?

(12, 18)

(16, 20)

(17, 24)

(20, 30)

17 is prime; 24 = 2³ × 3 → no overlap → coprime.

Explanation

17 is prime; 24 = 2³ × 3 → no overlap → coprime.

15) If GCD(25, x) = 1, which x works?

5

10

14

20

25 = 5²; 14 = 2 × 7 → no common factor → coprime.

Explanation

25 = 5²; 14 = 2 × 7 → no common factor → coprime.

16) If (a, b) = 1, what is always true?

A divides b

B divides a

There exist integers x, y with ax + by = 1

A = b

If (a, b) = 1 → there exist integers x, y with ax + by = 1.

Explanation

If (a, b) = 1 → there exist integers x, y with ax + by = 1.

17) Which of these pairs is coprime?

(8, 20)

(9, 27)

(11, 28)

(16, 40)

11 = prime; 28 = 2² × 7 → no shared factors → coprime.

Explanation

11 = prime; 28 = 2² × 7 → no shared factors → coprime.

18) If GCD(12, x) = 1, which x values make (12, x) coprime?

6

9

11

18

12 = 2² × 3; 11 is prime and doesn’t share 2 or 3 → coprime.

Explanation

12 = 2² × 3; 11 is prime and doesn’t share 2 or 3 → coprime.

19) Which pair is coprime?

(18, 27)

(25, 35)

(28, 42)

(29, 40)

29 = prime; 40 = 2³ × 5 → no common factors → coprime.

Explanation

29 = prime; 40 = 2³ × 5 → no common factors → coprime.

20) If GCD(a, b) = 1, what is always true about fractions a/b?

A/b is always > 1

A/b is never simplified

A/b is in simplest form

A/b is always a whole number

No further reduction is possible because they share no common divisor.

Explanation

No further reduction is possible because they share no common divisor.

Coprime Numbers in Algebraic Contexts

1) Solve for x: GCD(7, x) = 1. Which x works?

2) If a and b are coprime, then ax + by = 1 has:

3) Solve: If GCD(15, x) = 1, which value of x works?

4) Which pair is not coprime?

5) Solve: If GCD(21, x) = 1, which value of x works?

6) Which of these pairs are coprime?

7) Solve: GCD(18, x) = 1. Which x works?

8) If a and b are coprime, what is LCM(a, b)?

9) If a and b are coprime, then GCD(a², b²) = ?

10) Which of these pairs is not coprime?

11) Which statement is true?

12) Which pair is coprime?

13) If GCD(a, b) = 1, which of these equations has integer solutions?

14) Which pair is coprime?

15) If GCD(25, x) = 1, which x works?

16) If (a, b) = 1, what is always true?

17) Which of these pairs is coprime?

18) If GCD(12, x) = 1, which x values make (12, x) coprime?

19) Which pair is coprime?

20) If GCD(a, b) = 1, what is always true about fractions a/b?