Math: Number Theory: GCD, LCM, and Modular Arithmetic Worksheet

Question 1

Find the greatest common divisor (GCD) of 24 and 36. Show one method you used.

Accepted Answer

The GCD of 24 and 36 is 12. The factors of 24 include 1, 2, 3, 4, 6, 8, 12, and 24, and the factors of 36 include 1, 2, 3, 4, 6, 9, 12, 18, and 36, so the greatest common factor is 12.

Question 2

Find the least common multiple (LCM) of 6 and 15.

Accepted Answer

The LCM of 6 and 15 is 30. The multiples of 6 include 6, 12, 18, 24, and 30, and the multiples of 15 include 15 and 30, so the least common multiple is 30.

Question 3

Use prime factorization to find the GCD and LCM of 18 and 48.

Accepted Answer

The prime factorization of 18 is 2 x 3 x 3, and the prime factorization of 48 is 2 x 2 x 2 x 2 x 3. The GCD is 2 x 3 = 6, and the LCM is 2 x 2 x 2 x 2 x 3 x 3 = 144.

Question 4

A teacher has 32 pencils and 40 erasers. She wants to make identical supply bags with no pencils or erasers left over. What is the greatest number of supply bags she can make, and how many pencils and erasers will be in each bag?

Accepted Answer

The greatest number of supply bags is 8 because the GCD of 32 and 40 is 8. Each bag will have 4 pencils and 5 erasers.

Question 5

Two blinking lights flash at the same time. One flashes every 8 seconds, and the other flashes every 12 seconds. In how many seconds will they flash together again?

Accepted Answer

They will flash together again in 24 seconds. The LCM of 8 and 12 is 24, so both lights flash at the same time every 24 seconds.

Question 6

Find the remainder when 47 is divided by 5. Write your answer using modular notation in the form 47 ≡ r mod 5.

Accepted Answer

When 47 is divided by 5, the remainder is 2 because 5 x 9 = 45 and 47 - 45 = 2. In modular notation, 47 ≡ 2 mod 5.

Question 7

Find the value of 38 mod 7.

Accepted Answer

The value of 38 mod 7 is 3. This is because 7 x 5 = 35, and 38 - 35 = 3.

Question 8

A clock shows 9:00. What time will it show 17 hours later? Explain using modular arithmetic.

Accepted Answer

The clock will show 2:00. Since 9 + 17 = 26, and 26 mod 12 = 2, the hour shown will be 2.

Question 9

Complete the statement: 29 ≡ ___ mod 6. Explain your answer.

Accepted Answer

The completed statement is 29 ≡ 5 mod 6. This is because 6 x 4 = 24, and 29 - 24 = 5.

Question 10

Which numbers from this list are congruent to 2 mod 5: 7, 10, 12, 17, 21, 27? Explain how you know.

Accepted Answer

The numbers congruent to 2 mod 5 are 7, 12, 17, and 27. Each of these numbers leaves a remainder of 2 when divided by 5.

Question 11

Find the GCD and LCM of 45 and 60.

Accepted Answer

The GCD of 45 and 60 is 15. The LCM of 45 and 60 is 180 because 45 = 3 x 3 x 5 and 60 = 2 x 2 x 3 x 5, so the LCM is 2 x 2 x 3 x 3 x 5 = 180.

Question 12

A number leaves a remainder of 1 when divided by 4. Which of these could be the number: 18, 21, 25, 30, 33? List all possible answers and explain.

Accepted Answer

The possible numbers are 21, 25, and 33. Each of these numbers has a remainder of 1 when divided by 4.

Math: Number Theory: GCD, LCM, and Modular Arithmetic

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