cadaVR anatomy by Mozaik Education

Common multiples, lowest common multiple

Example 1

In a wildlife park, the caretakers feed two animal cubs. The monkey needs food every \latex{ 4 } \latex{ hours }, and the lion cub needs food every \latex{ 6 } \latex{ hours }. How many \latex{ hours } will pass before both animals are fed together at the same time if they have just been fed?

Solution

Let's represent the feedings with red and blue markings on a number line. The red markings are multiples of \latex{ 6 }, and the blue markings are multiples of \latex{ 4 }.

The points marked by both colours represent the common multiples of \latex{ 4 } and \latex{ 6}: \latex{ 0 }; \latex{ 12 }; \latex{ 24 }; \latex{ 36 }; ...

NOW

\latex{ (hours) }

\latex{ 0 }

\latex{ 4 }

\latex{ 6 }

\latex{ 8 }

\latex{ 12 }

\latex{ 16 }

\latex{ 18 }

\latex{ 20}

\latex{ 24}

\latex{ 28}

\latex{ 30}

\latex{ 32}

\latex{ 36}

The animals will be fed together in \latex{ 12}; \latex{24}; \latex{36}; ... \latex{ hours. }

In the example above, we were looking for natural numbers that are multiples of both \latex{ 4 } and \latex{ 6 }. These numbers are called common multiples. The common multiples of \latex{ 4 } and \latex{ 6 } are \latex{0}; \latex{12}; \latex{24}; \latex{36}; \latex{48}; \latex{60};... Among these common multiples, the smallest positive one is \latex{ 12 }. This is called the lowest common multiple of \latex{ 4 } and \latex{ 6 }.

Its notation is:

LCM of \latex{4} and \latex{6} = \latex{12}

The common multiples of two numbers are the numbers that are multiples of both numbers. The smallest positive common multiple is called the lowest common multiple.

We can find the lowest common multiple of two numbers by using prime factorisation.

The multiples of \latex{ 4 }: \latex{\textcolor{#0099ff}{2 \times 2} \times 0}; \latex{\textcolor{#0099ff}{2 \times 2} \times 1}; \latex{\textcolor{#0099ff}{2 \times 2} \times 2}; ...
The multiples of \latex{ 6 }: \latex{\textcolor{#0099ff}{2 \times 3} \times 0}; \latex{\textcolor{#0099ff}{2 \times 3} \times 1}; \latex{\textcolor{#0099ff}{2 \times 3} \times 2}; ...

The prime factorisation of the lowest common multiple of \latex{ 4 } and \latex{ 6 } must include the factors \latex{ 2 \;×\; 2 } and \latex{ 2 \;× 3\; }, as these are multiples of both \latex{ 4 } and \latex{ 6 }.

\latex{4 = 2 \times 2}

\latex{6 = 2 \times 3}

LCM of \latex{4} and \latex{6}\latex{= 2 \times 2 \times 3 = 12}

LCM of \latex{4} and \latex{6} \latex{= 2 \times 6 = 4 \times 3 = 12}

Example 2

What is the lowest common multiple of \latex{ 42 } and \latex{ 90 ?}

Solution

Let's break down the numbers into their prime factors:

\latex{42=2\times3\times7}
\latex{90=2\times3\times3\times5}

Both \latex{2 \times 3 \times 7} and \latex{2 \times 3 \times 3 \times 5} are included in the prime factorisation of the lowest common multiple of \latex{ 42 } and \latex{ 90 }.

\latex{2 \times 3 \times 7}

\latex{7 \times 2 \times 3}

\latex{2 \times 3 \times 3 \times 5}

\latex{\times 3 \times 5}

\latex{42 =}

\latex{90 =}

LCM of \latex{42} and \latex{90 =}

\latex{= 630}

The lowest common multiple of \latex{ 42 } and \latex{ 90 } is \latex{ 630 }.

The lowest common multiple is easy to calculate by arranging the prime factors in the following way:

LCM of \latex{42} and \latex{90 = 7 \times 90 = 630} or LCM of \latex{42} and \latex{90 = 42 \times 15 = 630}.

We can also use the lowest common multiple to find the common denominator of fractions.

Example 3

Add the fractions \latex{\frac{7}{60}} and \latex{\frac{5}{72}}.

Solution

First, bring the fractions to a common denominator.

The common denominator can be any common multiple of \latex{ 60 } and \latex{ 72 }, but it is best to choose the lowest common multiple.

\latex{60=2\times2\times3\times5}

\latex{72=2\times2\times2\times3\times3}

LCM of \latex{60} and \latex{72=}

\latex{=2\times 2\times 2\times 3\times 3\times 5=360}

As \latex{ 360 } is the lowest common multiple, it is best to use it as the common denominator.

\latex{\frac{7}{60} + \frac{5}{72} = \frac{7 \times 6}{360} + \frac{5 \times 5}{360} = \frac{42}{360} + \frac{25}{360} = \frac{67}{360}}

\latex{\times 2 \times 3}

\latex{\times 5}

Exercises

{{exercise_number}}. Draw a set diagram of the multiples of \latex{ 12 } and \latex{ 15 }. Write the two-digit multiples of \latex{ 12 } and \latex{ 15 } in the appropriate sets.

{{exercise_number}}. Draw a set diagram of the multiples of \latex{ 4, 6, } and \latex{ 10 }. Write the two-digit multiples of these numbers in the appropriate sets.

{{exercise_number}}. Which of the following numbers is a multiple of \latex{3 \times 3 \times 7?}

\latex{3 \times 3 \times 5 \times 7}

\latex{2 \times 3 \times 7}

\latex{2 \times 2 \times 3 \times 3 \times 7}

\latex{3 \times 3 \times 5 \times 11}

\latex{2 \times 3 \times 3 \times 5 \times 7}

\latex{2 \times 3 \times 3 \times 7 \times 11}

{{exercise_number}}. By how many times is \latex{A = 2 \times 2 \times 3 \times 3 \times 5 \times 7} greater than:

\latex{12};

\latex{35};

\latex{140?}

{{exercise_number}}. Two sailors, who are good friends, work on different cargo ships. Both ships leave Amsterdam at the same time but follow different routes. One ship completes its journey and returns to Amsterdam every \latex{ 18 } \latex{ days }, while the other returns every \latex{ 21 } \latex{ days }. How many \latex{ days } will it take for the two friends to meet again in Amsterdam?

{{exercise_number}}. A bicycle has \latex{ 35 } teeth on the front sprocket and \latex{ 21 } teeth on the rear sprocket. We mark one tooth on each sprocket. How many times does the pedal need to be turned for the marked teeth to return to their original position?

{{exercise_number}}. Write the lowest common multiple of numbers \latex{ A } and \latex{ B } in prime factor form.

\latex{A = 2 \times 2 \times 3 \times 5 \times 19}

\latex{B = 2 \times 3 \times 3 \times 5 \times 5 \times 7}

\latex{A = 2 \times 7 \times 7 \times 23}

\latex{B = 2 \times 2 \times 2 \times 7 \times 19}

\latex{A = 5 \times 7 \times 7 \times 7 \times 11}

\latex{B = 7 \times 7 \times 11 \times 11 \times 13 }

{{exercise_number}}. What is the lowest common multiple of \latex{ 42 }; \latex{ 60 } and \latex{ 18 }?

{{exercise_number}}. Calculate the lowest common multiple of the following numbers.

\latex{105} and \latex{90}

\latex{360} and \latex{108}

\latex{98} and \latex{84}

{{exercise_number}}. Determine the lowest common multiple and the highest common factor of the following pairs of numbers.

\latex{ 8 } and \latex{ 9 }

\latex{ 7 } and \latex{ 10 }

\latex{ 10 } and \latex{ 11 }

\latex{ 19 } and \latex{ 20 }

What do you notice?

{{exercise_number}}. Complete the additions and subtractions.

\latex{\frac{5}{8}+\frac{2}{15}}

\latex{\frac{7}{4}-\frac{3}{5}}

\latex{\frac{13}{14}+\frac{11}{19}}

\latex{\frac{3}{75}-\frac{2}{125}}

{{exercise_number}}. What is the product of the highest common factor and the lowest common multiple of the following pairs of numbers?

\latex{ 6 } and \latex{ 8 }

\latex{ 6 } and \latex{ 9 }

\latex{ 10 } and \latex{ 15 }

\latex{ 14 } and \latex{ 21 }

What do you notice?

{{exercise_number}}. In the prime factorisation of the number \latex{ A }, how many factors of \latex{ 2 } are needed, and in the prime factorisation of the number \latex{ B }, how many factors of \latex{ 3 } are needed to ensure that the lowest common multiple of the two numbers is \latex{2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 11}?

The lowest common multiple of \latex{A} and \latex{B} is \latex{2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 11}.

\latex{A = \underbrace{2 \times \dots \times 2}_{\text{x number of them}} \times 3 \times 3 \hspace{30 pt} B = 2 \times 2 \times \underbrace{3 \times \dots \times 3}_{\text{y number of them}} \times 11}

Quiz

Two natural numbers have a highest common factor of \latex{ 15 }, and the lowest common multiple is \latex{ 180 }. One of the numbers is \latex{ 45 }. What is the other number?

Mathematics 6.

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