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This is a simple question testing the student on their ability to calculate the lowest common multiple of two integers which are:

Part a) - coprime;

Part b) - where the greatest common divisor between the two integers is greater than one and not equal to either given number; and

Part c) - where one of the integer is a multiple of the other.

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a)

Here are the times tables for $\\var{a}$ and $\\var{b}$.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

$\\var{a}$

$\\var{2a}$

$\\var{3a}$

$\\var{4a}$

$\\var{5a}$

$\\var{6a}$

$\\var{7a}$

$\\var{8a}$

$\\var{9a}$

$\\var{10a}$

$\\var{11a}$

$\\var{12a}$

$\\var{13a}$

$\\var{b}$

$\\var{2b}$

$\\var{3b}$

$\\var{4b}$

$\\var{5b}$

$\\var{6b}$

$\\var{7b}$

$\\var{8b}$

$\\var{9b}$

$\\var{10b}$

$\\var{11b}$

$\\var{12b}$

$\\var{13b}$

The first number which appears in both lists is $\\var{a*b}$.

Alternately, notice that $\\var{a}$ and $\\var{b}$ don't have any factors in common, so their greatest common divisor is $1$. So the lowest common multiple is just the product of the two numbers.

b)

The lowest common multiple of $\\var{f}$ and $\\var{g}$ will be the product of the two numbers, divided by the greatest common divisor.

The greatest common divisor of $\\var{f}$ and $\\var{g}$ is $\\var{gcd_fg}$.

Therefore, the lowest common multiple will is

\\[\\frac{\\var{f}\\times\\var{g}}{\\var{gcd_fg}}=\\var{lcm_fg}\\text{.}\\]

c)

$\\var{d}$ is a multiple of $\\var{c}$, as $\\var{d/c}\\times\\var{c}=\\var{d}.$

The lowest common multiple of $\\var{c}$ and $\\var{d}$ will therefore be $\\var{d/c} \\times \\var{c} = 1 \\times \\var{d} = \\var{d}$.

", "statement": "

The lowest common multiple of two numbers is the first number which appears in both numbers' times tables.

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What is the lowest common multiple of $\\var{a}$ and $\\var{b}$?

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What is the lowest common multiple of $\\var{f}$ and $\\var{g}$?

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What is the lowest common multiple of $\\var{c}$ and $\\var{d}$?

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