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This question tests the student's ability to identify the factors of some composite numbers and the highest common factors of two numbers.

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i)

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$\\var{fourfac}$ has four factors: $1$, $3$, $\\var{fourfac/3}$ and $\\var{fourfac}$.

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It is possible to pair the factors up to prove that they are factors.

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\\\begin{align} 1\\times\\var{fourfac}&=\\var{fourfac}\\text{.}\\\\ 3\\times\\var{fourfac/3}&=\\var{fourfac}\\text{.}\\\\ \\end{align} \

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ii)

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$\\var{sixfac}$ has six factors: $1$, $2$, $3$, $\\var{sixfac/3}$, $\\var{sixfac/2}$ and $\\var{sixfac}$.

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Again, it is possible to pair the factors up to prove that they are factors.

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\\\begin{align} 1\\times\\var{sixfac}&=\\var{sixfac}\\text{.}\\\\ 2\\times\\var{sixfac/2}&=\\var{sixfac}\\text{.}\\\\ 3\\times\\var{sixfac/3}&=\\var{sixfac}\\text{.}\\\\ \\end{align} \

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#### b)

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We now look for common factors between the two lists of factors, and the highest common factor will be the largest of these.

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For $\\var{fourfac}$ and $\\var{sixfac}$, the highest common factor is $\\var{hc}$.

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#### c)

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Dividing both the numerator and denominator by the highest common factor gives:

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\$\\frac{\\var{sixfac}}{\\var{fourfac}} = \\frac{\\frac{\\var{sixfac}}{\\var{hc}}}{\\frac{\\var{fourfac}}{\\var{hc}}} = \\frac{\\var{sixfac/hc}}{\\var{fourfac/hc}}\\text{.}\$

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Simplify $\\var{sixfac}/\\var{fourfac}$ by finding the highest common factor.

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Identify the factors of the following numbers in ascending order.

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i)     Factors of $\\var{fourfac}$:            $1$, [[4]], [[5]], $\\var{fourfac}$

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ii)     Factors of $\\var{sixfac}$:          [[0]], $2$, [[1]], [[2]], $\\var{sixfac/2}$, [[3]]

What is the heighest common factor of $\\var{fourfac}$ and $\\var{sixfac}$?

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The heighest common factor is [[0]]

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If the same number appears in a list of factors for two numbers, it is a common factor. The largest of these common factors is the highest common factor.

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Use the result above to reduce the following fraction to its simplest form.

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$\\displaystyle \\frac{\\var{sixfac}}{\\var{fourfac}} =$ [[0]] [[1]]

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To simplify the fraction, divide both the numerator and denominator by the highest common factor.

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