Coprime Numbers: Recognition, Properties, and Applications

1) 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.

2) 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.

3) 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.

4) 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.

5) 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.

6) 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.

7) 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.

8) 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.

9) 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.

10) 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.

11) 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.”

12) 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.

13) 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.

14) 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.

15) 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.

16) 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.

17) 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.

18) 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.

19) 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.

20) 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.

Identifying and Applying Coprime Numbers

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

2)

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

2) Which of these pairs is coprime?

3) Which is not a coprime pair?

4) Which pair is coprime?

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

6) Which pair is not coprime?

7) Which consecutive numbers are coprime?

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

9) Which pair is coprime?

10) Which of the following pairs is coprime?

11) Which statement about coprime numbers is true?

12) Which is a coprime pair?

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

14) Which number is coprime to 35?

15) Which of these sets are all coprime pairs?

16) Which pair is not coprime?

17) Which of these pairs are coprime?

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

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

20) Which of the following pairs is coprime?