Coprime Numbers: Recognition, Properties, and Applications

1) Which pair is coprime even though one number is composite?

(9, 12)

(15, 20)

(9, 16)

(18, 27)

9 = 3² and 16 = 2⁴. Different prime factors but both composite → coprime.

Explanation

9 = 3² and 16 = 2⁴. Different prime factors but both composite → coprime.

2) Which is a coprime pair?

(12, 18)

(8, 15)

(10, 20)

(16, 24)

8 = 2³; 15 = 3 × 5 → no shared factors → coprime.

Explanation

8 = 2³; 15 = 3 × 5 → no shared factors → coprime.

3) Which of the following pairs is coprime?

(8, 12)

(15, 25)

(14, 28)

(14, 25)

14 = 2 × 7 and 25 = 5 × 5. They share no common factors, so they’re coprime.

Explanation

14 = 2 × 7 and 25 = 5 × 5. They share no common factors, so they’re coprime.

4) Which pair is not coprime?

(13, 20)

(16, 24)

(9, 22)

(17, 30)

16 = 2⁴, 24 = 2³ × 3. They share 2’s, so they’re not coprime.

Explanation

16 = 2⁴, 24 = 2³ × 3. They share 2’s, so they’re not coprime.

5) Which of these sets are all coprime pairs?

(9, 20), (11, 30), (15, 21)

(7, 25), (9, 22), (11, 28)

(12, 20), (18, 27), (25, 35)

(14, 28), (20, 30), (16, 40)

Each pair has no shared factor — they’re all coprime.

Explanation

Each pair has no shared factor — they’re all coprime.

6) Which number is coprime to 35?

15

21

22

28

35 = 5 × 7; 22 = 2 × 11. No shared factors → coprime.

Explanation

35 = 5 × 7; 22 = 2 × 11. No shared factors → coprime.

7) If GCD(21, x) = 1, which x value makes them coprime?

10

14

21

28

21 = 3 × 7, and 10 = 2 × 5. No shared factors, so they’re coprime.

Explanation

21 = 3 × 7, and 10 = 2 × 5. No shared factors, so they’re coprime.

8) If two numbers are coprime, their GCD is always:

Their product

0

Their sum

1

That’s the definition — their only common factor is 1.

Explanation

That’s the definition — their only common factor is 1.

9) Which of these pairs are coprime?

(4, 9)

(6, 15)

(12, 25)

Both (4, 9) & (12, 25)

Both pairs share no common primes → both are coprime.

Explanation

Both pairs share no common primes → both are coprime.

10) Which statement about coprime numbers is true?

Both must be prime

One must be prime and one composite

They share no common factors except 1

They must be consecutive numbers

True statement about coprime numbers → “They share no common factors except 1.”

Explanation

True statement about coprime numbers → “They share no common factors except 1.”

11) If numbers a and b are coprime, which is always true?

LCM(a, b) = a + b

LCM(a, b) = a × b

A – b = 1

A and b are both prime

They have no overlap in factors, so multiply directly for the LCM.

Explanation

They have no overlap in factors, so multiply directly for the LCM.

12) Which of the following pairs is coprime?

(18, 27)

(11, 18)

(20, 24)

(21, 28)

11 = prime; 18 = 2 × 3². No common factors → coprime.

Explanation

11 = prime; 18 = 2 × 3². No common factors → coprime.

13) Which pair is coprime?

(25, 40)

(18, 24)

(28, 49)

(19, 42)

19 = prime; 42 = 2 × 3 × 7 → no shared factors → coprime.

Explanation

19 = prime; 42 = 2 × 3 × 7 → no shared factors → coprime.

14) If GCD(14, n) = 1, then n could be:

7

14

28

9

14 = 2 × 7; 9 = 3² → no shared primes → coprime.

Explanation

14 = 2 × 7; 9 = 3² → no shared primes → coprime.

15) Which consecutive numbers are coprime?

18 and 19

20 and 21

21 and 22

All of the above

Consecutive numbers always differ by 1 → no shared factors.

Explanation

Consecutive numbers always differ by 1 → no shared factors.

16) Which pair is not coprime?

(11, 12)

(15, 45)

(13, 40)

(17, 38)

15 = 3 × 5; 45 = 3² × 5 → many common factors.

Explanation

15 = 3 × 5; 45 = 3² × 5 → many common factors.

17) If GCD(a, b) = 1, what can you say about a and b?

They are multiples

They are coprime

They must be prime

They are both odd

If GCD(a, b) = 1 → They are coprime.

Explanation

If GCD(a, b) = 1 → They are coprime.

18) Which pair is coprime?

(20, 28)

(9, 28)

(15, 25)

(16, 32)

9 = 3²; 28 = 2² × 7 → no shared factors.

Explanation

9 = 3²; 28 = 2² × 7 → no shared factors.

19) Which is not a coprime pair?

(3, 10)

(4, 15)

(8, 12)

(7, 20)

Both have 2’s as factors → not coprime.

Explanation

Both have 2’s as factors → not coprime.

20) Which of these pairs is coprime?

(5, 12)

(6, 13)

(8, 15)

All of the above

(5,12), (6,13), (8,15) each have no shared factors.

Explanation

(5,12), (6,13), (8,15) each have no shared factors.

Identifying and Applying Coprime Numbers

1) Which pair is coprime even though one number is composite?

2)

What first name or nickname would you like us to use?

2) Which is a coprime pair?

3) Which of the following pairs is coprime?

4) Which pair is not coprime?

5) Which of these sets are all coprime pairs?

6) Which number is coprime to 35?

7) If GCD(21, x) = 1, which x value makes them coprime?

8) If two numbers are coprime, their GCD is always:

9) Which of these pairs are coprime?

10) Which statement about coprime numbers is true?

11) If numbers a and b are coprime, which is always true?

12) Which of the following pairs is coprime?

13) Which pair is coprime?

14) If GCD(14, n) = 1, then n could be:

15) Which consecutive numbers are coprime?

16) Which pair is not coprime?

17) If GCD(a, b) = 1, what can you say about a and b?

18) Which pair is coprime?

19) Which is not a coprime pair?

20) Which of these pairs is coprime?