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guided practice find the least common multiple (lcm). 1. 4, 7 2. 14, 21…

Question

guided practice
find the least common multiple (lcm).

  1. 4, 7
  2. 14, 21, 28
  3. 4, 8, 12, 10
  4. 30, 48
  5. 3, 9, 15
  6. 10, 40, 50
  7. jerry and his dad are walking around the track. jerry completes one lap every 8 minutes. his dad completes one lap every 6 minutes. they start together. in how many minutes will they be together at the starting line again?

independent practice
find the least common multiple (lcm).

  1. 6, 9
  2. 8, 12
  3. 15, 20
  4. 6, 14
  5. 18, 27
  6. 8, 10, 12
  7. 12, 15, 22

Explanation:

Response
  1. For 4, 7:
  • # Explanation:
  • ## Step1: Prime - factorize the numbers
  • Prime - factorize \(4 = 2\times2\) and \(7\) is a prime number, so \(7=7\).
  • ## Step2: Calculate the LCM
  • The LCM is the product of the highest - power prime factors. Here, the LCM\((4,7)=2\times2\times7 = 28\).
  • # Answer:
  • 28
  1. For 14, 21, 28:
  • # Explanation:
  • ## Step1: Prime - factorize the numbers
  • \(14 = 2\times7\), \(21=3\times7\), \(28 = 2\times2\times7\).
  • ## Step2: Determine the highest - power prime factors
  • For prime factor 2, the highest power is \(2^2\); for prime factor 3, the highest power is \(3^1\); for prime factor 7, the highest power is \(7^1\).
  • ## Step3: Calculate the LCM
  • \(LCM(14,21,28)=2^2\times3\times7=84\).
  • # Answer:
  • 84
  1. For 4, 8, 12, 16:
  • # Explanation:
  • ## Step1: Prime - factorize the numbers
  • \(4 = 2\times2\), \(8 = 2\times2\times2\), \(12=2\times2\times3\), \(16 = 2\times2\times2\times2\).
  • ## Step2: Determine the highest - power prime factors
  • For prime factor 2, the highest power is \(2^4\); for prime factor 3, the highest power is \(3^1\).
  • ## Step3: Calculate the LCM
  • \(LCM(4,8,12,16)=2^4\times3 = 48\).
  • # Answer:
  • 48
  1. For 30, 48:
  • # Explanation:
  • ## Step1: Prime - factorize the numbers
  • \(30=2\times3\times5\), \(48 = 2\times2\times2\times2\times3\).
  • ## Step2: Determine the highest - power prime factors
  • For prime factor 2, the highest power is \(2^4\); for prime factor 3, the highest power is \(3^1\); for prime factor 5, the highest power is \(5^1\).
  • ## Step3: Calculate the LCM
  • \(LCM(30,48)=2^4\times3\times5=240\).
  • # Answer:
  • 240
  1. For 3, 9, 15:
  • # Explanation:
  • ## Step1: Prime - factorize the numbers
  • \(3 = 3\), \(9 = 3\times3\), \(15=3\times5\).
  • ## Step2: Determine the highest - power prime factors
  • For prime factor 3, the highest power is \(3^2\); for prime factor 5, the highest power is \(5^1\).
  • ## Step3: Calculate the LCM
  • \(LCM(3,9,15)=3^2\times5 = 45\).
  • # Answer:
  • 45
  1. For 10, 40, 50:
  • # Explanation:
  • ## Step1: Prime - factorize the numbers
  • \(10 = 2\times5\), \(40=2\times2\times2\times5\), \(50 = 2\times5\times5\).
  • ## Step2: Determine the highest - power prime factors
  • For prime factor 2, the highest power is \(2^3\); for prime factor 5, the highest power is \(5^2\).
  • ## Step3: Calculate the LCM
  • \(LCM(10,40,50)=2^3\times5^2=200\).
  • # Answer:
  • 200
  1. For Jerry and his dad:
  • Jerry completes one lap every 8 minutes and his dad completes one lap every 6 minutes.
  • # Explanation:
  • ## Step1: Prime - factorize the numbers
  • \(6 = 2\times3\), \(8 = 2\times2\times2\).
  • ## Step2: Determine the highest - power prime factors
  • For prime factor 2, the highest power is \(2^3\); for prime factor 3, the highest power is \(3^1\).
  • ## Step3: Calculate the LCM
  • \(LCM(6,8)=2^3\times3=24\).
  • # Answer:
  • 24 minutes
  1. For 6, 9:
  • # Explanation:
  • ## Step1: Prime - factorize the numbers
  • \(6 = 2\times3\), \(9 = 3\times3\).
  • ## Step2: Determine the highest - power prime factors
  • For prime factor 2, the highest power is \(2^1\); for prime factor 3, the highest power is \(3^2\).
  • ## Step3: Calculate the LCM
  • \(LCM(6,9)=2\times3^2 = 18\).
  • # Answer:…

Answer:

  1. For 4, 7:
  • # Explanation:
  • ## Step1: Prime - factorize the numbers
  • Prime - factorize \(4 = 2\times2\) and \(7\) is a prime number, so \(7=7\).
  • ## Step2: Calculate the LCM
  • The LCM is the product of the highest - power prime factors. Here, the LCM\((4,7)=2\times2\times7 = 28\).
  • # Answer:
  • 28
  1. For 14, 21, 28:
  • # Explanation:
  • ## Step1: Prime - factorize the numbers
  • \(14 = 2\times7\), \(21=3\times7\), \(28 = 2\times2\times7\).
  • ## Step2: Determine the highest - power prime factors
  • For prime factor 2, the highest power is \(2^2\); for prime factor 3, the highest power is \(3^1\); for prime factor 7, the highest power is \(7^1\).
  • ## Step3: Calculate the LCM
  • \(LCM(14,21,28)=2^2\times3\times7=84\).
  • # Answer:
  • 84
  1. For 4, 8, 12, 16:
  • # Explanation:
  • ## Step1: Prime - factorize the numbers
  • \(4 = 2\times2\), \(8 = 2\times2\times2\), \(12=2\times2\times3\), \(16 = 2\times2\times2\times2\).
  • ## Step2: Determine the highest - power prime factors
  • For prime factor 2, the highest power is \(2^4\); for prime factor 3, the highest power is \(3^1\).
  • ## Step3: Calculate the LCM
  • \(LCM(4,8,12,16)=2^4\times3 = 48\).
  • # Answer:
  • 48
  1. For 30, 48:
  • # Explanation:
  • ## Step1: Prime - factorize the numbers
  • \(30=2\times3\times5\), \(48 = 2\times2\times2\times2\times3\).
  • ## Step2: Determine the highest - power prime factors
  • For prime factor 2, the highest power is \(2^4\); for prime factor 3, the highest power is \(3^1\); for prime factor 5, the highest power is \(5^1\).
  • ## Step3: Calculate the LCM
  • \(LCM(30,48)=2^4\times3\times5=240\).
  • # Answer:
  • 240
  1. For 3, 9, 15:
  • # Explanation:
  • ## Step1: Prime - factorize the numbers
  • \(3 = 3\), \(9 = 3\times3\), \(15=3\times5\).
  • ## Step2: Determine the highest - power prime factors
  • For prime factor 3, the highest power is \(3^2\); for prime factor 5, the highest power is \(5^1\).
  • ## Step3: Calculate the LCM
  • \(LCM(3,9,15)=3^2\times5 = 45\).
  • # Answer:
  • 45
  1. For 10, 40, 50:
  • # Explanation:
  • ## Step1: Prime - factorize the numbers
  • \(10 = 2\times5\), \(40=2\times2\times2\times5\), \(50 = 2\times5\times5\).
  • ## Step2: Determine the highest - power prime factors
  • For prime factor 2, the highest power is \(2^3\); for prime factor 5, the highest power is \(5^2\).
  • ## Step3: Calculate the LCM
  • \(LCM(10,40,50)=2^3\times5^2=200\).
  • # Answer:
  • 200
  1. For Jerry and his dad:
  • Jerry completes one lap every 8 minutes and his dad completes one lap every 6 minutes.
  • # Explanation:
  • ## Step1: Prime - factorize the numbers
  • \(6 = 2\times3\), \(8 = 2\times2\times2\).
  • ## Step2: Determine the highest - power prime factors
  • For prime factor 2, the highest power is \(2^3\); for prime factor 3, the highest power is \(3^1\).
  • ## Step3: Calculate the LCM
  • \(LCM(6,8)=2^3\times3=24\).
  • # Answer:
  • 24 minutes
  1. For 6, 9:
  • # Explanation:
  • ## Step1: Prime - factorize the numbers
  • \(6 = 2\times3\), \(9 = 3\times3\).
  • ## Step2: Determine the highest - power prime factors
  • For prime factor 2, the highest power is \(2^1\); for prime factor 3, the highest power is \(3^2\).
  • ## Step3: Calculate the LCM
  • \(LCM(6,9)=2\times3^2 = 18\).
  • # Answer:
  • 18
  1. For 8, 12:
  • # Explanation:
  • ## Step1: Prime - factorize the numbers
  • \(8 = 2\times2\times2\), \(12=2\times2\times3\).
  • ## Step2: Determine the highest - power prime factors
  • For prime factor 2, the highest power is \(2^3\); for prime factor 3, the highest power is \(3^1\).
  • ## Step3: Calculate the LCM
  • \(LCM(8,12)=2^3\times3=24\).
  • # Answer:
  • 24
  1. For 15, 20:
  • # Explanation:
  • ## Step1: Prime - factorize the numbers
  • \(15 = 3\times5\), \(20=2\times2\times5\).
  • ## Step2: Determine the highest - power prime factors
  • For prime factor 2, the highest power is \(2^2\); for prime factor 3, the highest power is \(3^1\); for prime factor 5, the highest power is \(5^1\).
  • ## Step3: Calculate the LCM
  • \(LCM(15,20)=2^2\times3\times5 = 60\).
  • # Answer:
  • 60
  1. For 6, 14:
  • # Explanation:
  • ## Step1: Prime - factorize the numbers
  • \(6 = 2\times3\), \(14=2\times7\).
  • ## Step2: Determine the highest - power prime factors
  • For prime factor 2, the highest power is \(2^1\); for prime factor 3, the highest power is \(3^1\); for prime factor 7, the highest power is \(7^1\).
  • ## Step3: Calculate the LCM
  • \(LCM(6,14)=2\times3\times7 = 42\).
  • # Answer:
  • 42
  1. For 18, 27:
  • # Explanation:
  • ## Step1: Prime - factorize the numbers
  • \(18 = 2\times3\times3\), \(27=3\times3\times3\).
  • ## Step2: Determine the highest - power prime factors
  • For prime factor 2, the highest power is \(2^1\); for prime factor 3, the highest power is \(3^3\).
  • ## Step3: Calculate the LCM
  • \(LCM(18,27)=2\times3^3=54\).
  • # Answer:
  • 54
  1. For 8, 10, 12:
  • # Explanation:
  • ## Step1: Prime - factorize the numbers
  • \(8 = 2\times2\times2\), \(10=2\times5\), \(12=2\times2\times3\).
  • ## Step2: Determine the highest - power prime factors
  • For prime factor 2, the highest power is \(2^3\); for prime factor 3, the highest power is \(3^1\); for prime factor 5, the highest power is \(5^1\).
  • ## Step3: Calculate the LCM
  • \(LCM(8,10,12)=2^3\times3\times5=120\).
  • # Answer:
  • 120