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Factor and multiple
Write down \latex{ 15 } as the product of two natural numbers.
\latex{15=3\times5}
\latex{15=1\times15}
\latex{ 3 } is a
factor of \latex{ 15 }.
\latex{ 15 } is a
multiple of \latex{ 3 }.
factor of \latex{ 15 }.
\latex{ 15 } is a
multiple of \latex{ 3 }.
\latex{ 5 } is a
factor of \latex{ 15 }.
\latex{ 15 } is a
multiple of \latex{ 5 }.
factor of \latex{ 15 }.
\latex{ 15 } is a
multiple of \latex{ 5 }.
\latex{ 1 } is a
factor of \latex{ 15 }.
\latex{ 15 } is a
multiple of \latex{ 1 }.
factor of \latex{ 15 }.
\latex{ 15 } is a
multiple of \latex{ 1 }.
\latex{ 15 } is a
factor of \latex{ 15 }.
\latex{ 15 } is a
multiple of \latex{ 15 }.
factor of \latex{ 15 }.
\latex{ 15 } is a
multiple of \latex{ 15 }.
The factors of \latex{ 15 } are \latex{ 1 }; \latex{ 3 }; \latex{ 5 } and \latex{ 15 }.
The product of some pairs of factors is the number itself.
Example 1
List the factors of the following numbers: \latex{ 6 }; \latex{ 8 }; \latex{ 2 }; \latex{ 1 }; \latex{ 0 }.
Solution
\latex{6=\textcolor{#e3004f}{1}\times\textcolor{#009ee0}{6}=2\times3,} | the factors of \latex{ 6 } are: \latex{ 1 }; \latex{ 2 }; \latex{ 3 }; \latex{ 6 }. |
\latex{8=\textcolor{#e3004f}{1}\times\textcolor{#009ee0}{8}=2\times4,} | the factors of \latex{ 8 } are: \latex{ 1 }; \latex{ 2 }; \latex{ 4 }; \latex{ 8 }. |
\latex{2=\textcolor{#e3004f}{1}\times\textcolor{#009ee0}{2},} | the factors of \latex{ 2 } are: \latex{ 1 }; \latex{ 2 }. |
\latex{1=\textcolor{#e3004f}{1}\times\textcolor{#009ee0}{1},} | the factors of \latex{ 1 } is: \latex{ 1 } |
\latex{0\times0=0\times1=0\times2=..,} | the factors of \latex{ 0 } are: \latex{ 0 }; \latex{ 1 }; \latex{ 2 }; \latex{ 3 }; \latex{ 4 }; \latex{ 5 }; \latex{ 6 }... |
\latex{ 1 } is a factor of every number and every number is a factor of itself.
Every factor of a number can be found with the help of factor pairs.
Example 2
Find the factors of \latex{ 54 }.
Solution
Write down the factors of \latex{ 54 } using factor pairs.
Since \latex{7 \times 7 = 49 \lt 54 \;and \;8 \times 8 = 64 \gt 54},
the smaller factor of the pair cannot be greater than \latex{ 7 }.
Look at the numbers from \latex{ 1 } to \latex{ 7 } to decide whether they are factors of \latex{ 54 }.
The factors of \latex{ 54 }: \latex{ 1 }; \latex{ 2 }; \latex{ 3 }; \latex{ 6 }; \latex{ 9 }; \latex{ 18 }; \latex{ 27 } and \latex{ 54 }.
the smaller factor of the pair cannot be greater than \latex{ 7 }.
Look at the numbers from \latex{ 1 } to \latex{ 7 } to decide whether they are factors of \latex{ 54 }.
The factors of \latex{ 54 }: \latex{ 1 }; \latex{ 2 }; \latex{ 3 }; \latex{ 6 }; \latex{ 9 }; \latex{ 18 }; \latex{ 27 } and \latex{ 54 }.
\latex{ 1\times54 }
\latex{ 2\times27 }
\latex{ 3\times18 }
\latex{ 2\times27 }
\latex{ 3\times18 }
\latex{ 6\times9 }
Example 3
List the multiples of \latex{ 8 }; \latex{ 3 }; \latex{ 1 } and \latex{ 0 }.
Solution
multiples of \latex{ 8 }: \latex{ 0 }; \latex{ 8 }; \latex{ 16 }; \latex{ 24 }; \latex{ 32 }; ... | multiples of \latex{ 1 }: \latex{ 0 }; \latex{ 1 }; \latex{ 2 }; \latex{ 3 }; \latex{ 4 }; ... |
multiples of \latex{ 3 }: \latex{ 0 }; \latex{ 3 }; \latex{ 6 }; \latex{ 9 }; \latex{ 12 }; ... | multiple of \latex{ 0 }: \latex{ 0 } |
Every natural number is a multiple of \latex{ 1 }.
\latex{ 0 } is a multiple of every natural number.
Every natural number is a multiple of itself.
\latex{ 0 } is a multiple of every natural number.
Every natural number is a multiple of itself.
Exercises
{{exercise_number}}. Are the following statements true or false?
a) \latex{ 45 } is a factor of zero.
c) Zero is a multiple of \latex{ 45 }.
e) Zero is a factor of \latex{ 45 }.
g) \latex{ 45 } is a factor of \latex{ 1 }.
c) Zero is a multiple of \latex{ 45 }.
e) Zero is a factor of \latex{ 45 }.
g) \latex{ 45 } is a factor of \latex{ 1 }.
i) \latex{ 1 } is a factor of \latex{ 0 }.
b) \latex{ 45 } is a factor of \latex{ 45 }.
d) \latex{ 45 } is a multiple of itself.
f) \latex{ 45 } is a multiple of zero.
h) \latex{ 1 } is a factor of \latex{ 45 }.
d) \latex{ 45 } is a multiple of itself.
f) \latex{ 45 } is a multiple of zero.
h) \latex{ 1 } is a factor of \latex{ 45 }.
\latex{ 1 }; \latex{ 12 }
HELP
Factor pairs of \latex{ 12 }:
\latex{ 2 }; \latex{ 6 }
\latex{ 3 }; \latex{ 4 }
\latex{ 3 }; \latex{ 4 }
{{exercise_number}}. List the first five multiples of the following numbers.
a) \latex{ 5 }
b) \latex{ 25 }
c) \latex{ 7 }
d) \latex{ 21 }
e) \latex{ 9 }
f) \latex{ 19 }
{{exercise_number}}. List all the factors of the following numbers.
a) \latex{ 8 }
b) \latex{ 18 }
c) \latex{ 24 }
d) \latex{ 72 }
e) \latex{ 96 }
{{exercise_number}}. A frog jumps \latex{ 50\,cm }, while a grasshopper jumps \latex{ 80\,cm }. How far do they get with \latex{ 3 }; \latex{ 7 }; \latex{ 10 } and \latex{ 25 } jumps? If they jump in the same direction, how far is the first point where they meet again? (A number line can help in solving this exercise.)
{{exercise_number}}. \latex{ 60 } is a multiple of which numbers?
{{exercise_number}}. Perform the following divisions, then check your results.
\latex{325 \div 13 =}
\latex{7,200 \div 228 =}
\latex{22,317 \div 43 =}
a) List at least four factors of each dividend.
b) Which factor is a factor of at least two of the dividends?
c) Which number is a factor of every natural number?
{{exercise_number}}. List numbers that have exactly
a) \latex{ 2 } factors;
b) \latex{ 3 } factors;
c) \latex{ 4 } factors.
Quiz
How many digits does the number have whose fourfold can be created by moving the digit \latex{ 8 } from the last place to the first?
...
\latex{ 2\;\,\;8 \, \times\, 4 }
\latex{ 1\;\,2 }
\latex{ 8 }
...
