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In this video, we’ll explore how to use different strategies to find prime factorization using exponents.
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This is sometimes called writing a number as a product of its prime factors and can be hugely useful in helping us to calculate the greatest common factor or least common multiple of two or more numbers.
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Let’s begin by recalling what we mean by a factor.
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A factor is a number that divides into another number without leaving a remainder.
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When listing factors of a number, we often list them in pairs for ease.
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For example, the product of one and 12 is 12.
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So a factor pair of 12 is one and 12.
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We also know that two times six is 12 and three times four is 12.
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So all the factors of 12 are one, 12, two, six, three, and four.
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And what about a prime number?
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A prime number is a number that has exactly two factors, one and itself.
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The first few prime numbers that you should be able to recognize are two, three, five, seven, 11, 13, 17, and 19.
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Now, a common mistake is to think that nine is a prime number.
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But its factors are one, three, and nine.
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It has three factors, not two.
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Similarly, one is not a prime number, since it only has one factor, itself.
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When we write a number as a product of its prime factors, that’s called prime factorization, we look to find all the prime numbers that multiply together to make the original number.
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There are a couple of methods.
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We can draw a prime factor tree or use the division method.
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We’ll begin by taking a look at the factor tree method.
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Which of the following is the prime factorization of 18?
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Is it (A) 18, (B) two times three times three, (C) two, (D) two times nine, or (E) three?
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When we find the prime factorization of 18, we’re looking for all the prime numbers which multiply together to make 18.
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And we know the first few prime numbers to be two, three, five, seven, 11, and 13.
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It can be nice to begin by testing whether our smallest prime number, whether two, is a factor of the original number.
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Well, 18 is an even number.
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So we know two is definitely a factor.
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In fact, 18 divided by two is nine.
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And our factor tree begins.
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Its first two branches are two and nine.
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Now, since two is a prime number, we circle it.
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And this indicates we stop at this branch here.
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Nine, however, is not a prime number.
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So we repeat this process for nine.
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Nine is not even, so it’s not divisible by two.
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And instead, we go to the next prime on our list.
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Three is a factor of nine.
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In fact, three squared is nine.
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So nine divided by three is three.
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This means the next two branches on our tree are three and three.
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Three is prime, so we circle both of these.
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And we see that we can go no further.
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We aren’t done, though.
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We know that these three prime numbers multiply together to make the original.
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So in other words, 18 is the product of them.
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It’s two times three times three, which is option (B).
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Notice that we can also write this as two times three squared.
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Had the question asked us to write our original number in exponent form, then this is what we would have needed to have done.
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Be careful, though, a common mistake is just to think that we can give our answers in a list, that’s two, three, three, or as a sum, two plus three plus three.
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Well, both of these are incorrect answers.
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We must give the solution as a product.
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In our next example, we’ll consider an alternative method.
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It’s called the division method.
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Write 60 as a product of its prime factors, giving the answer in exponent form.
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Writing a number as a product of its prime factors is sometimes called prime factorization.
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Product means time.
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So we want all the prime numbers which multiply together to make here the original number that’s 60.
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We’re going to use the division method.
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Now, the first step in this method is to find the smallest prime number that’s also a factor of the original number, so a factor of 60.
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Our first few primes are two, three, five, seven, and 11.
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Now, since 60 is an even number, we know that it must be divisible by two.
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It has a factor of two.
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And so we divide 60 by this number.
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60 divided by two is 30.
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That’s equivalent to saying 60 is equal to two times 30.
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Our next step is really similar.
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This time, though, we divide the quotient by the smallest prime number that’s a factor of it.
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Once again, 30 is an even number, so we can divide it by two.
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And when we do, we find that 30 divided by two is 15.
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This is equivalent to saying that 30 is equal to two times 15.
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So we replace 30 with two times 15.
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And we see that we’ve now written 60 as two times two times 15.
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We continue to repeat this process until our quotient, the number we get after dividing, is a prime number.
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15 is not divisible by two, so we move to the next smallest prime number.
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That’s three.
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And when we divide 15 by three, we get five.
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Now, that is a prime number.
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So we finished.
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We replace 15 with its prime factors, with three and five.
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And we find that 60 is two times two times three times five.
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We’re not quite done, though.
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The question asks us to write this in exponent form.
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Another way of writing two times two is writing two squared.
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Remember, when we square a number, we times it by itself.
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And so in exponent form, 60 is two squared times three times five.
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In our next example, we’ll look at the prime factorization of a much larger number, this time going back to using a factor tree.
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Write the prime factorization of 392 in exponent form.
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The prime factorization of 392 is the product of all the prime numbers, which multiply together to make 392.
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And it’s always sensible to list out the first few prime numbers.
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They are two, three, five, seven, 11, and 13.
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We’re going to draw a prime factor tree to work out the prime factorization of 392.
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We begin by finding a factor of 392.
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A really easy one is to check whether the number is even.
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If it’s even, then it has a factor of two.
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392 is even, which means we can divide it by two.
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And when we do, we get a result of 196.
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This means that, on our factor tree, the first two branches are two and 196.
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Remember, though, we said two is a prime number.
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That means we circle the number two.
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And it indicates to us that on this branch at least, we stop here.
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Next, we look for a factor of 196.
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Well, 196 is also even.
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So it must have a factor of two.
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In fact, when we divide 196 by two, we get 98.
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And that means the next two branches in our tree are two and 98.
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We pop a circle around this two to show that it’s the end of this branch, and we carry on with the 98.
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98 can also be divided by two to give us 49.
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And so our next two branches are two and 49.
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Circling the two, we look for a factor of 49.
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We know 49 in fact is divisible by seven.
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And since 49 divided by seven is seven, our next two branches are both seven.
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And since seven is a prime number, we circle both of these.
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And we’re done with our factor tree.
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We now know that 392 is the product of all these prime numbers, everything in the circle.
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It’s two times two times two times seven times seven.
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But the question told us to give our answer in exponent form.
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Two times two times two is two cubed, and seven times seven is seven squared.
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So 392 can be written as two cubed times seven squared.
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Now, it’s important to realize that had we chosen a different starting factor of 392, we would’ve still achieved the same answer.
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For example, imagine we’d first spotted that 392 is the same as 49 times eight.
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Neither of these are prime numbers, so we’re going to perform the usual process on both branches.
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We write 49 as seven times seven, which means that these two branches both have a seven.
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Seven is a prime, so we circle them and finish here.
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We might then spot that eight is the same as four times two.
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We circle the two, but four is not prime, so we continue.
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Four can be written as two times two.
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So our next two branches are both two, and we circle them because they’re prime and finish here.
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Once again, we have two times two times two times seven times seven.
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Let’s now have a look at a prime factorization of a larger number using the division method.
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Find the prime factorization of 792.
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We’re going to use the division method to find the prime factorization of this number.
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When we perform this process, we want to find all the prime numbers that multiply together to make 792.
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And so we list the first few prime numbers.
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And then the first step is to find the smallest prime number that’s also a factor of the original number.
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792 is even, so we know it has a factor of two.
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Let’s divide then 792 by two.
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792 divided by two is 396.
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Now, we can equivalently say that 792 is equal to two times 396.
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Step two is to repeat this process with the quotient, the number we got when we performed the division.
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So we need to find the smallest prime factor of 396.
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Well, once again, it’s even, so that’s two.
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We’re, therefore, going to calculate 396 divided by two.
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That’s 198.
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This means 396 is equal to two times 198.
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And it means, in turn, we can rewrite 792 as two times two times 198.
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We continue to divide our quotients by their smallest prime factor.
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And we only stop when our quotient is also prime.
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So we need to divide 198 by its smallest prime factor, which is once again two.
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198 divided by two is 99.
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So equivalently, we can say that 198 must be equal to two times 99.
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We now need to divide 99 by its smallest prime factor.
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But 99 is not even.
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So it’s not divisible by two.
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We do, however, know that it’s divisible by three.
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And of course, we might recall that we can check for divisibility by three by adding the digits of the number together.
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If their sum is divisible by three, then the original number is also divisible by three.
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In this case, nine plus nine is 18.
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Now, 18 is divisible by three.
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So 99 must be divisible by three.
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And in fact, when we divide 99 by three, we get 33.
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So we rewrite 99 is three times 33.
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And 792 is now two times two times two times three times 33.
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The smallest prime factor of 33, the smallest factor that’s also a prime number, is three.
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And when we divide 33 by three, we get 11.
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Now, 11 itself is also a prime number.
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It’s in our list here, so we finish dividing.
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We write 33 is three times 11, and this means 792 is two times two times two times three times three times 11.
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We can go even further and write this in exponent form.
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Two times two times two is two cubed and three times three is three squared.
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We can use a dot instead of a multiplication symbol, and we see that 792 is two cubed times three squared times 11.
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In this video, we’ve learned that writing a number as a product of its prime factors, also called prime factorization, is finding all the prime numbers that multiply together to make the original number.
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We also learned that we absolutely must write these as a product, not a list or a sum.
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So 30 is two times three times five.
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We also saw that we have a couple of methods that we can use, and these are called the prime factor tree method or the division method.