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The Venn diagram shows the prime factors of x and 30. a) Find the value of: i) x ii) n. b) Find the G.C.F of 30 and X.

Mathematics

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3 parts
a(i)
216
a(ii)
5
b
6

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Here are the solutions to your questions:

Part 29

The Venn diagram shows the prime factors of xx and 30.
The left circle represents PFx_x (Prime Factors of xx).
The right circle represents PF30_{30} (Prime Factors of 30).

From the Venn diagram:

  • Factors unique to xx (left-only part of PFx_x): 222^2, 323^2
  • Common factors (intersection): 212^1, 313^1
  • Factors unique to 30 (right-only part of PF30_{30}): nn

Part a(i)216

x

Find the value of

  1. To find xx, multiply all the prime factors shown in the circle for PFx_x.

    This includes the factors unique to xx and the common factors.

    x=(22×32)×(21×31)x = (2^2 \times 3^2) \times (2^1 \times 3^1)

  2. Combine the powers of the same prime factors.

    x=22+1×32+1x = 2^{2+1} \times 3^{2+1} x=23×33x = 2^3 \times 3^3

  3. Calculate the value.

    x=8×27x = 8 \times 27 x=216x = 216

    The value of xx is 216\boxed{216}.

Part a(ii)5

n

  1. We know that 30=2×3×530 = 2 \times 3 \times 5.

    From the Venn diagram, the prime factors of 30 are the common factors and the factors unique to 30.

    30=(21×31)×n30 = (2^1 \times 3^1) \times n

  2. Simplify the common factors.

    30=(2×3)×n30 = (2 \times 3) \times n 30=6×n30 = 6 \times n

  3. Solve for nn.

    n=306n = \frac{30}{6} n=5n = 5

    The value of nn is 5\boxed{5}.

Part b6

Find the G.C.F of 30 and X.

  1. The G.C.F (Greatest Common Factor) is the product of the prime factors found in the intersection of the Venn diagram.

    G.C.F(30,x)=21×31\text{G.C.F}(30, x) = 2^1 \times 3^1

  2. Calculate the product.

    G.C.F(30,x)=2×3\text{G.C.F}(30, x) = 2 \times 3 G.C.F(30,x)=6\text{G.C.F}(30, x) = 6

    The G.C.F of 30 and xx is 6\boxed{6}.

Part c· 1 step

Calculate the L.C.M of 30 and X using the Venn diagram.

  1. The L

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