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Writing composite numbers as the product of prime numbers
\latex{ 2 }
\latex{ 12 }
\latex{ \times }
\latex{ 6 }
\latex{ 2 }
\latex{ 2 }
\latex{ 2 }
\latex{ 6 }
\latex{ 3 }
\latex{ 2 }
\latex{ \times }
\latex{ \times }
Frank invented a clever machine. When he puts a composite number into the machine, it divides it into the product of two factors. One of the factors (or both) will be a prime number. What can the machine be used for?
Write \latex{ 6 } and \latex{ 21 } as the product of two natural numbers greater than \latex{ 1 }.
\latex{6=2\times 3} \latex{21=3\times 7}
Both factors are prime numbers.
Write \latex{ 36 } as the product of two natural numbers greater than \latex{ 1 }. If one of the factors is a composite number, keep doing the same until both factors become prime numbers.
\latex{ 36 } is now expressed as the product of its prime factors.
\latex{36}
\latex{36}
\latex{36}
\latex{36}
\latex{2}
\latex{2}
\latex{2}
\latex{2}
\latex{2}
\latex{2}
\latex{2}
\latex{2}
\latex{3}
\latex{3}
\latex{3}
\latex{3}
\latex{3}
\latex{3}
\latex{3}
\latex{3}
\latex{18}
\latex{12}
\latex{4}
\latex{9}
\latex{6}
\latex{6}
\latex{9}
\latex{4}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{36=2\times 2\times 3\times 3}
\latex{36=3\times3\times 2\times 2}
\latex{36=2\times 2\times 3\times 3}
\latex{36=2\times 3\times 2\times 3}
No matter how you divide \latex{ 36 }, the prime factors are the same. Arrange these prime factors in a non-descending order.
\latex{36=2\times 2\times 3\times 3}
Write \latex{ 90 } as the product of its prime factors.
\latex{90}
\latex{2}
\latex{2}
\latex{2}
\latex{3}
\latex{3}
\latex{3}
\latex{3}
\latex{3}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{90}
\latex{90}
\latex{90}
\latex{90}
\latex{90}
\latex{90}
\latex{45}
\latex{45}
\latex{30}
\latex{30}
\latex{30}
\latex{10}
\latex{9}
\latex{15}
\latex{6}
\latex{15}
\latex{9}
\latex{6}
\latex{10}
\latex{15}
\latex{3}
\latex{3}
\latex{3}
\latex{3}
\latex{3}
\latex{3}
\latex{3}
\latex{3}
\latex{3}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{2}
\latex{2}
\latex{2}
\latex{2}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{\times}
\latex{5}
\latex{5}
\latex{5}
\latex{5}
\latex{5}
\latex{5}
\latex{5}
\latex{90=2\times 3\times 3\times 5}
\latex{90=2\times 5\times 3\times 3}
\latex{90=3\times 5\times 2\times 3}
\latex{90=3\times 3\times 2\times 5}
\latex{90=3\times 2\times 3\times 5}
\latex{90=2\times 5\times 3\times 3}
\latex{90=3\times 5\times 2\times 3}
In the end, you get the same prime factors again. Arrange these in a non-descending order.
\latex{90=2\times 3\times 3\times 5}
Every composite number can be expressed as the product of its prime factors. This decomposition is unique apart from the order of the factors.
A method for finding the prime factors:
\latex{36}
\latex{18}
\latex{9}
\latex{3}
\latex{1}
\latex{2}
\latex{2}
\latex{3}
\latex{3}
\latex{90}
\latex{45}
\latex{15}
\latex{5}
\latex{1}
\latex{2}
\latex{3}
\latex{3}
\latex{5}
\latex{90=2\times 3\times 3\times 5}
\latex{36=2\times 2\times 3\times 3}
Identical prime factors can be written in index form as well:
\latex{3\times 3=3^{2} .}
\latex{ 3 } is the base: the repeating factor;
\latex{ 2 } is the exponent: the number of times the factor is repeated.
You should read it as: \latex{ 3 } raised to the second power.
\latex{ 36 } and \latex{ 90 } written in index form: \latex{36=2^{2}\times 3^{2};\;90=2\times 3^{2}\times 5.}
\latex{90}
\latex{45}
\latex{15}
\latex{5}
\latex{1}
\latex{2}
\latex{3}
\latex{3}
\latex{5}
end
factor pair
factor pair
factor pair
factor pair
prime factor
prime factor
prime factor
prime factor
prime factors
Using prime factors to list the divisors of a number
Example 1
Write down all the factors of \latex{ 90 }.
Solution 1
Use factor pairs to list the divisors of \latex{ 90 }.
\latex{1}
\latex{2}
\latex{3}
\latex{5}
\latex{6}
\latex{9}
\latex{10}
\latex{15}
\latex{18}
\latex{30}
\latex{45}
\latex{90}
Solution 2
The divisors can also be determined using prime factors.
\latex{90=\underbrace{2\times 3\times 3\times 5}_{\text{product}}}
Arrange the divisors based on the number of prime factors they include:
Factors of \latex{ 90 }
Composing factors
All numbers are divisible by \latex{1}.
Prime factors.
Product of two primes.
Product of three primes.
Product of four primes.
\latex{1}
\latex{2;3;5}
\latex{\underbrace{2\times 3;}_{\text{\textcolor{e51f52}{6}}}}
\latex{\underbrace{2\times 5;}_{\text{\textcolor{e51f52}{10}}}}
\latex{\underbrace{3\times 3;}_{\text{\textcolor{e51f52}{9}}}}
\latex{\underbrace{3\times 5}_{\text{\textcolor{e51f52}{15}}}}
\latex{\underbrace{2\times 3\times3;}_{\text{\textcolor{e51f52}{18}}}}
\latex{\underbrace{2\times 3\times5;}_{\text{\textcolor{e51f52}{30}}}}
\latex{\underbrace{3\times 3\times5}_{\text{\textcolor{e51f52}{45}}}}
\latex{\underbrace{2\times 3\times3\times5}_{\text{\textcolor{e51f52}{90}}}}
\latex{ 90 } has \latex{ 12 } factors.
Example 2
Is a) \latex{42}; b) \latex{39}; c) \latex{63} a factor of \latex{294}?
Solution
Divide the number into prime factors and write it as a product of its prime factors.
\latex{294}
\latex{147}
\latex{49}
\latex{7}
\latex{1}
\latex{2}
\latex{3}
\latex{7}
\latex{7}
\latex{294=2\times 3\times 7\times 7}
\latex{42}
\latex{21}
\latex{7}
\latex{1}
\latex{2}
\latex{3}
\latex{7}
\latex{42=\underbrace{2\times 3\times 7}_{\text{}}}
\latex{294=\overbrace{2\times 3\times 7}^{\text{}}\times 7}
\latex{42}
All the prime factors of \latex{ 42 } are also prime factors of \latex{ 294 }; therefore, \latex{ 42 } is a factor of \latex{ 294 }.
\latex{39}
\latex{13}
\latex{1}
\latex{3}
\latex{13}
\latex{39=3\times13}
One of the prime factors of \latex{ 39 } (\latex{ 13 }) is not a prime factor of \latex{ 294 }; therefore, \latex{ 39 } is not a factor of \latex{ 294 }.
\latex{63}
\latex{21}
\latex{7}
\latex{3}
\latex{3}
\latex{63=3\times3\times7}
\latex{1}
\latex{7}
\latex{ 3 } appears twice among the prime factors of \latex{ 63 }; therefore, \latex{ 63 } is not a factor of \latex{ 294 }.
Exercises
{{exercise_number}}. Write the following numbers as a product of prime factors, then determine all their factors.
- \latex{60}
- \latex{84}
- \latex{252}
- \latex{678}
{{exercise_number}}. List all the three-digit numbers that can be obtained using number cards \latex{2} \latex{4} and \latex{6} . Then write them as a product of their prime factors.
{{exercise_number}}. Arrange \latex{ 256;\, 385;\, 407 } and \latex{ 441 } in ascending order according to the number of their factors.
{{exercise_number}}. \latex{ 36 } congruent squares are used to make various rectangles. How many different rectangles can be made using every square? Which one of these has the smallest circumference?
{{exercise_number}}. \latex{ 24 } congruent cubes are used to make various (non-congruent) cuboids. How large is their surface area if you must use all the cubes? (Two cuboids are congruent if the lengths of their corresponding sides are the same.).
{{exercise_number}}. Without calculating, determine how many times greater the product of \latex{11\times 13\times 2\times 5} is than
- \latex{11};
- \latex{13};
- \latex{13\times5};
- \latex{11\times13};
- \latex{13\times2\times5};
- \latex{11\times13\times2\times5};
- \latex{22};
- \latex{26};
- \latex{55};
- \latex{110}.
{{exercise_number}}. Multiply the prime numbers on the cards below to create the largest number possible, then determine all of its factors.
\latex{2} \latex{2} \latex{2} \latex{3} \latex{7}
{{exercise_number}}. Use the prime factors of \latex{ 5;\, 21;\, 35;\, 45;\, 75 } and \latex{ 125 } to determine which of these numbers is a factor of \latex{ 525 }.
{{exercise_number}}. Use the prime factors of the numbers \latex{ 7 }; \latex{ 12 }; \latex{ 21 }; \latex{ 33 }; \latex{ 91 }; \latex{ 108 } and \latex{ 198 } to determine whether \latex{ 1,386 } is a multiple of the given number.
{{exercise_number}}. Zack multiplied his age, his shoe size and the number of his siblings. He got \latex{154}. What is Zack’s shoe size?
{{exercise_number}}. Erika bought a new phone. She can remember her new seven-digit phone number the following way: the first two digits are the largest prime number less than \latex{ 50 }, the next two digits are the smallest prime number greater than \latex{ 50 }, and the last three digits are the product of the first four prime numbers. What is Erika's phone number?
Quiz
The product of the captain's age, the number of his children, and the length of his boat (in \latex{metres}) is \latex{ 10,509 }. (Every number is a whole number.) How old is the captain, and how long is his boat?
