Math: Number Theory: GCD, LCM, and Modular Arithmetic
Finding common factors, common multiples, and remainders
Math: Number Theory: GCD, LCM, and Modular Arithmetic
Finding common factors, common multiples, and remainders
Math - Grade 6-8
- 1
Find the greatest common divisor (GCD) of 24 and 36. Show one method you used.
List the factors of each number and look for the greatest factor they share.
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. - 2
Find the least common multiple (LCM) of 6 and 15.
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. - 3
Use prime factorization to find the GCD and LCM of 18 and 48.
For the GCD, use only the shared prime factors. For the LCM, use all prime factors with the greatest number of times each appears.
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. - 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?
The greatest number of identical groups is found using the GCD.
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. - 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?
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. - 6
Find the remainder when 47 is divided by 5. Write your answer using modular notation in the form 47 ≡ r mod 5.
Find the largest multiple of 5 that is less than or equal to 47.
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. - 7
Find the value of 38 mod 7.
The value of 38 mod 7 is 3. This is because 7 x 5 = 35, and 38 - 35 = 3. - 8
A clock shows 9:00. What time will it show 17 hours later? Explain using modular arithmetic.
On a 12-hour clock, hours wrap around after 12.
The clock will show 2:00. Since 9 + 17 = 26, and 26 mod 12 = 2, the hour shown will be 2. - 9
Complete the statement: 29 ≡ ___ mod 6. Explain your answer.
The completed statement is 29 ≡ 5 mod 6. This is because 6 x 4 = 24, and 29 - 24 = 5. - 10
Which numbers from this list are congruent to 2 mod 5: 7, 10, 12, 17, 21, 27? Explain how you know.
Divide each number by 5 and check the remainder.
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. - 11
Find the GCD and LCM of 45 and 60.
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. - 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.
Numbers that are 1 more than a multiple of 4 have remainder 1 when divided by 4.
The possible numbers are 21, 25, and 33. Each of these numbers has a remainder of 1 when divided by 4.