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) 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.

4) 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.

5) 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.

6) 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.

7) 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.

8) 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.

9) 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.

10) 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.

11) 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.

12) 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.

13) 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.

14) 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.

15) 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.

16) 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.

17) 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.

18) 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.

19) 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.

20) 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.

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) Which of these pairs is coprime?

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

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

6) Which pair is not coprime?

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

8) Which of these pairs are coprime?

9) Which pair is coprime?

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

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

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

13) Which pair is coprime?

14) Which statement is true?

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

16) Which of these pairs is not coprime?

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

18) Which pair is coprime?

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

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