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Find the LCM of the following. Leave the answers as prime factors.

Mathematics

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5 parts
a

22×32×52^2 \times 3^2 \times 5

b

23×32×52^3 \times 3^2 \times 5

c

24×32×522^4 \times 3^2 \times 5^2

d

2×32×52×72 \times 3^2 \times 5^2 \times 7

e

2×33×52×72 \times 3^3 \times 5^2 \times 7

Step-by-step solution

15 steps

To find the Least Common Multiple (LCM) of numbers expressed in prime factor form, we first express each number in index notation. Then, we identify all unique prime factors present in any of the numbers and take the highest power for each of those unique prime factors. Finally, we multiply these highest powers together.

Part a· 3 steps

The given numbers are: 2×2×32 \times 2 \times 3; 2×3×3×52 \times 3 \times 3 \times 5; 2×2×52 \times 2 \times 5

  1. Express each number in index notation.

    N1=22×31N_1 = 2^2 \times 3^1 N2=21×32×51N_2 = 2^1 \times 3^2 \times 5^1 N3=22×51N_3 = 2^2 \times 5^1
  2. Identify all unique prime factors and their highest powers.

    Unique prime factors are 2, 3, and 5.
    Highest power of 2: 222^2 (from N1N_1 and N3N_3)
    Highest power of 3: 323^2 (from N2N_2)
    Highest power of 5: 515^1 (from N2N_2 and N3N_3)

  3. Multiply the highest powers of the unique prime factors.

    LCM=22×32×51\text{LCM} = 2^2 \times 3^2 \times 5^1

    The LCM for part a is 22×32×5\boxed{2^2 \times 3^2 \times 5}.

Part b

23×32×52^3 \times 3^2 \times 5

The given numbers are: 2×3×32 \times 3 \times 3; 2×2×2×32 \times 2 \times 2 \times 3; 2×2×3×52 \times 2 \times 3 \times 5

  1. Express each number in index notation.

    N1=21×32N_1 = 2^1 \times 3^2 N2=23×31N_2 = 2^3 \times 3^1 N3=22×31×51N_3 = 2^2 \times 3^1 \times 5^1
  2. Identify all unique prime factors and their highest powers.

    Unique prime factors are 2, 3, and 5.
    Highest power of 2: 232^3 (from N2N_2)
    Highest power of 3: 323^2 (from N1N_1)
    Highest power of 5: 515^1 (from N3N_3)

  3. Multiply the highest powers of the unique prime factors.

    LCM=23×32×51\text{LCM} = 2^3 \times 3^2 \times 5^1

    The LCM for part b is 23×32×5\boxed{2^3 \times 3^2 \times 5}.

Part c

24×32×522^4 \times 3^2 \times 5^2

The given numbers are: 2×2×2×2×32 \times 2 \times 2 \times 2 \times 3; 2×3×5×52 \times 3 \times 5 \times 5; 2×2×3×3×52 \times 2 \times 3 \times 3 \times 5

  1. Express each number in index notation.

    N1=24×31N_1 = 2^4 \times 3^1 N2=21×31×52N_2 = 2^1 \times 3^1 \times 5^2 N3=22×32×51N_3 = 2^2 \times 3^2 \times 5^1
  2. Identify all unique prime factors and their highest powers.

    Unique prime factors are 2, 3, and 5.
    Highest power of 2: 242^4 (from N1N_1)
    Highest power of 3: 323^2 (from N3N_3)
    Highest power of 5: 525^2 (from N2N_2)

  3. Multiply the highest powers of the unique prime factors.

    LCM=24×32×52\text{LCM} = 2^4 \times 3^2 \times 5^2

    The LCM for part c is 24×32×52\boxed{2^4 \times 3^2 \times 5^2}.

Part d

2×32×52×72 \times 3^2 \times 5^2 \times 7

The given numbers are: 3×3×53 \times 3 \times 5; 2×5×5×72 \times 5 \times 5 \times 7; 2×3×72 \times 3 \times 7; 3×5×73 \times 5 \times 7

  1. Express each number in index notation.

    N1=32×51N_1 = 3^2 \times 5^1 N2=21×52×71N_2 = 2^1 \times 5^2 \times 7^1 N3=21×31×71N_3 = 2^1 \times 3^1 \times 7^1 N4=31×51×71N_4 = 3^1 \times 5^1 \times 7^1
  2. Identify all unique prime factors and their highest powers.

    Unique prime factors are 2, 3, 5, and 7.
    Highest power of 2: 212^1 (from N2N_2 and N3N_3)
    Highest power of 3: 323^2 (from N1N_1)
    Highest power of 5: 525^2 (from N2N_2)
    Highest power of 7: 717^1 (from N2N_2, N3N_3, and N4N_4)

  3. Multiply the highest powers of the unique prime factors.

    LCM=21×32×52×71\text{LCM} = 2^1 \times 3^2 \times 5^2 \times 7^1

    The LCM for part d is 2×32×52×7\boxed{2 \times 3^2 \times 5^2 \times 7}.

Part e

2×33×52×72 \times 3^3 \times 5^2 \times 7

The given numbers are: 2×3×52 \times 3 \times 5; 3×5×73 \times 5 \times 7; 2×3×3×32 \times 3 \times 3 \times 3; 3×5×5×73 \times 5 \times 5 \times 7

  1. Express each number in index notation.

    N1=21×31×51N_1 = 2^1 \times 3^1 \times 5^1 N2=31×51×71N_2 = 3^1 \times 5^1 \times 7^1 N3=21×33N_3 = 2^1 \times 3^3 N4=31×52×71N_4 = 3^1 \times 5^2 \times 7^1
  2. Identify all unique prime factors and their highest powers.

    Unique prime factors are 2, 3, 5, and 7.
    Highest power of 2: 212^1 (from N1N_1 and N3N_3)
    Highest power of 3: 333^3 (from N3N_3)
    Highest power of 5: 525^2 (from N4N_4)
    Highest power of 7: 717^1 (from N2N_2 and N4N_4)

  3. Multiply the highest powers of the unique prime factors.

    LCM=21×33×52×71\text{LCM} = 2^1 \times 3^3 \times 5^2 \times 7^1

    The LCM for part e is 2×33×52×7\boxed{2 \times 3^3 \times 5^2 \times 7}.

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