Prime Factorization
Coach Rivera has $13$ players at practice and wants equal lines of more than one player. It can't be done: $13$ players fit only in one line of $13$, or $13$ lines of one. With $12$ players she has lots of choices: $2 \times 6$, $3 \times 4$, $4 \times 3$... What makes $12$ so flexible and $13$ so stubborn? The answer is hiding in their factors.
Prime and Composite Numbers
A prime can only be built one way. A composite can be split into smaller pieces. Those pieces are what we are going to hunt for.
Classify the following numbers in the table according to whether they are prime (they only have two factors: $1$ and themselves) or composite (they have more than two…
Classify the following numbers in the table according to whether they are prime (they only have two factors: $1$ and themselves) or composite (they have more than two factors):
$11, 14, 19, 21, 23, 27, 31, 33$
| Prime Numbers | Composite Numbers |
|---|---|
Factor Trees and the Ladder Method
Every composite number can be written as a product of primes. These primes are its "building blocks", and the process of finding them is called prime factorization. Better still, the answer is unique: no matter how you split a number, you always end with the same primes (just in a different order).
A factor tree splits a number into any two factors, then keeps splitting every composite factor until all the branches end in primes.
Another way is the ladder method (also called the vertical line method). Divide by the smallest prime that works, write it on the right, and keep dividing the quotient until you reach $1$.
Look at the following "division tower" (prime factorization) and identify the missing numbers in the empty gaps so that it is correct:
Look at the following "division tower" (prime factorization) and identify the missing numbers in the empty gaps so that it is correct:
Find the prime factorization of the number $40$ using the vertical line method and express the result as a product of powers.
Find the prime factorization of the number $40$ using the vertical line method and express the result as a product of powers.
Fill in the empty gaps in this factor tree for the number $120$. Remember that each branch must end in a prime number for the factorization to be complete. Then write…
Fill in the empty gaps in this factor tree for the number $120$. Remember that each branch must end in a prime number for the factorization to be complete. Then write the final result using powers.
Exponents and Shortcuts
When the same prime shows up several times, we write it with an exponent: $2 \cdot 2 \cdot 2 = 2^3$. The exponent counts how many times the prime is used. To turn a factorization back into the number, work out each power first, then multiply.
$$2^2 \cdot 3 \cdot 5 = 4 \cdot 3 \cdot 5 = 60$$Play with the powers of $2$ and see how fast they grow:
Numbers that end in zeros
Since $10 = 2 \cdot 5$, every zero at the end of a number gives you one $2$ and one $5$ for free. Split off the zeros first, then factor what is left.
Find out which number it is in each case by working out the powers and multiplying the factors: a) $2^2 \cdot 3 \cdot 5 = $ b) $2 \cdot 3^3 = $ c) $3^2 \cdot 11 = $
Find out which number it is in each case by working out the powers and multiplying the factors:
- a) $2^2 \cdot 3 \cdot 5 = $
- b) $2 \cdot 3^3 = $
- c) $3^2 \cdot 11 = $
To organize a video game tournament, we need to break down the number of participants into their "building blocks" (prime factors). Find the prime factorization using…
To organize a video game tournament, we need to break down the number of participants into their "building blocks" (prime factors). Find the prime factorization using the vertical line method for the following numbers:
- a) $126$
- b) $180$
- c) $420$
Sometimes, to factor numbers ending in zero, we can use mental "shortcuts". For example, we know that $10 = 2 \cdot 5$. Factor the following numbers using this trick to…
Sometimes, to factor numbers ending in zero, we can use mental "shortcuts". For example, we know that $10 = 2 \cdot 5$.
Factor the following numbers using this trick to save time on the vertical line:
- a) $1.200$
- b) $4.500$
- c) $18.000$
Reading a Factorization
A factorization tells you a lot about a number without multiplying anything. A prime factor of a number is a factor that is prime: these are exactly the primes in its factorization.
A few more things you can read off:
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$N$ is even if a $2$ appears in its factorization.
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$N$ is a multiple of $10$ if both a $2$ and a $5$ appear.
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Multiplying numbers adds the exponents of each prime. Dividing subtracts them.
Mystery-number puzzles
Clues like "multiple of $8$" or "still divisible by $5$ after dividing" are clues about exponents. "Multiple of $8 = 2^3$" means the exponent of $2$ is at least $3$. Dividing by a prime lowers its exponent by $1$.
Without doing the multiplication, look at the following factorization of a secret number $n$: $n = 2^2 \cdot 3 \cdot 5$ Indicate whether the following statements are…
Without doing the multiplication, look at the following factorization of a secret number $n$:
$n = 2^2 \cdot 3 \cdot 5$
Indicate whether the following statements are true or false, justifying your answer:
- a) The number $n$ is a multiple of $5$.
- b) The number $n$ is divisible by $7$.
- c) The number $3$ is a factor of $n$.
- d) The number $n$ is even.
Look at the factorization of a "mystery" number $N$: $N = 2^3 \cdot 3 \cdot 5^2$ Without needing to calculate the value of $N$, reason whether the following statements…
Look at the factorization of a "mystery" number $N$:
$N = 2^3 \cdot 3 \cdot 5^2$
Without needing to calculate the value of $N$, reason whether the following statements are true or false:
- a) $N$ is an even number.
- b) $N$ is divisible by $9$.
- c) $10$ is a factor of $N$.
- d) $N$ is a multiple of $25$.
A mystery number $n$ has the following prime factorization: $n = 2^a \cdot 3^b \cdot 5^c$. We know that: If we divide $n$ by $10$, the result is still divisible by $5$.…
A mystery number $n$ has the following prime factorization: $n = 2^a \cdot 3^b \cdot 5^c$.
We know that:
-
If we divide $n$ by $10$, the result is still divisible by $5$.
-
If we divide $n$ by $18$, the result is no longer divisible by $3$.
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The number $n$ is a multiple of $8$, but if we divide it by $2$, it stops being a multiple of $8$.
Calculate the values of the exponents $a$, $b$ and $c$ and find the value of the number $n$.
Finding All the Factors
Once you have the prime factorization, you can list every factor by combining the primes, with no trial and error. Take each prime and choose how many copies to use, from none up to its exponent.
For $45 = 3^2 \cdot 5$: the powers of $3$ are $1, 3, 9$ and the powers of $5$ are $1, 5$. Multiply every pair.
The factors of $45$ are $1, 3, 5, 9, 15, 45$. That is $6$ factors, which is a $3 \times 2$ grid.
Factor pairs are useful for equal groups. If a factor of $150$ tells you how many go in each box, the matching factor tells you how many boxes. Rectangles work the same way: a rectangle with area $N$ has side lengths that are a factor pair of $N$.
Lucy wants to organize her $28$ video games into identical boxes. For the organization to be "mathematically perfect", she has decided that the number of video games in…
Lucy wants to organize her $28$ video games into identical boxes. For the organization to be "mathematically perfect", she has decided that the number of video games in each box must be a prime factor of $28$.
In how many ways can she organize her boxes and how many games would go in each one? (Hint: start by finding the prime factorization of $28$).
A pastry chef has made $150$ artisan chocolates. He wants to store them in boxes so that each box has the same number of chocolates and none are left over. a) Find the…
A pastry chef has made $150$ artisan chocolates. He wants to store them in boxes so that each box has the same number of chocolates and none are left over.
- a) Find the prime factorization of $150$.
- b) Using the factors from the previous section, write down all the possible ways to pack the chocolates if each box must have between $10$ and $30$ units.
Find the smallest number that has exactly $6$ factors, knowing that only the first two prime numbers appear in its prime factorization. Explain your reasoning by…
Find the smallest number that has exactly $6$ factors, knowing that only the first two prime numbers appear in its prime factorization. Explain your reasoning by analyzing the exponents of its factorization.
GCF and LCM with Prime Factorization
Prime factorizations are the fastest way to find the GCF (greatest common factor) and the LCM (least common multiple) of big numbers. Some books write the GCF as GCD (greatest common divisor); it is the same thing.
Take $360 = 2^3 \cdot 3^2 \cdot 5$ and $300 = 2^2 \cdot 3 \cdot 5^2$.
The middle is the GCF: $2 \cdot 2 \cdot 3 \cdot 5 = 60$. Everything in the whole diagram is the LCM.
Use the GCF when you need to split things into the largest equal groups or pieces (for example, the biggest square tile or cube-shaped box that fits every length exactly; then divide each length by it to count how many fit). Use the LCM when things repeat on a schedule and you want to know when they line up again (for example, abilities that recharge every so many seconds).
A logistics company wants to manufacture the largest possible cubic boxes to transport three types of products that come in rectangular containers. The dimensions of the…
A logistics company wants to manufacture the largest possible cubic boxes to transport three types of products that come in rectangular containers. The dimensions of the products are:
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Product A: $360$ in
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Product B: $480$ in
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Product C: $540$ in
Use prime factorization to find the side length of the cubic box (which must be the Greatest Common Factor) and express this measurement as a product of its prime factors. How many boxes of that size will fit in total if we add those of the three products?
Three gamers are playing an online video game. Their characters have special abilities that recharge at different times: The first character launches an attack every…
Three gamers are playing an online video game. Their characters have special abilities that recharge at different times:
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The first character launches an attack every $2^3 \cdot 3$ seconds.
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The second character uses a shield every $2^2 \cdot 3^2$ seconds.
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The third character heals every $2 \cdot 3 \cdot 5$ seconds.
If at the start of the game at 18:00:00 all three have used their ability at the same time, how many seconds must pass for them to coincide again? Express the waiting time first as a prime factorization and then in minutes and seconds.
Given the numbers $A = 2^4 \cdot 3^2 \cdot 5 \cdot 11$ and $B = 2^3 \cdot 3^4 \cdot 7$. Without calculating the numerical value of $A$ or $B$, give a reasoned answer to:…
Given the numbers $A = 2^4 \cdot 3^2 \cdot 5 \cdot 11$ and $B = 2^3 \cdot 3^4 \cdot 7$.
Without calculating the numerical value of $A$ or $B$, give a reasoned answer to:
- a) Is the product $A \cdot B$ divisible by $55$?
- b) Is the product $A \cdot B$ a multiple of $54$?
- c) What is the prime factorization of the number that results from working out $M.C.D.(A, B)$?
Back to the start: Coach Rivera's $13$ players can only form one line because $13$ is prime: its only factors are $1$ and $13$. The number $12 = 2^2 \cdot 3$ is composite and has $(2 + 1)(1 + 1) = 6$ factors ($1, 2, 3, 4, 6, 12$), so she can make many equal groups.