Finding factors of a number | Factors and multiples | Pre-Algebra | Khan Academy

Summary

This script is a detailed explanation of finding all the factors of 120. It starts with the obvious factors, 1 and 120, and then systematically checks divisibility by 2, 3, 4, 5, 6, 8, 10, and other numbers up to 11. The video uses various divisibility rules, such as checking the sum of digits for 3 and ignoring the last digit for 4, to determine factors. It also demonstrates how to use long division to confirm divisibility. The factors found are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120, providing a comprehensive list of all numbers that 120 is divisible by.

Takeaways

• 🔢 The number 120 is divisible by 1 and itself, with 1 being the smallest factor and 120 being the largest.
• 👉 120 is an even number, hence divisible by 2, with 60 being the corresponding factor when divided by 2.
• 📐 The divisibility rule for 3 is satisfied if the sum of the digits is divisible by 3, and 120 meets this criterion with factors 3 and 40.
• 📘 To check divisibility by 4, look at the last two digits; since 20 is divisible by 4, 120 is too, with 30 being the factor.
• 💯 A number ending in 0 or 5 is divisible by 5, confirming 5 and 24 as factors of 120.
• 🔄 Being divisible by both 2 and 3 confirms divisibility by 6, resulting in factors 6 and 20.
• ❌ The number 7 does not divide evenly into 120, as shown by the long division process.
• 🔢 Divisibility by 8 is confirmed by the long division process, with 15 being the factor.
• ❌ The sum of the digits of 120 is 3, which is divisible by 3 but not by 9, thus 120 is not divisible by 9.
• 🔄 A number ending in 0 is divisible by 10, making 10 and 12 factors of 120.
• ❌ 11 does not divide evenly into 120, leaving a remainder, thus 11 is not a factor.

Q & A

What is the smallest factor of 120?

The smallest factor of 120 is 1, as every whole number is divisible by 1.

What is the largest factor of 120?

The largest factor of 120 is 120 itself, as no number larger than 120 can divide into 120 without a remainder.

Why is 2 a factor of 120?

2 is a factor of 120 because 120 is an even number, and all even numbers are divisible by 2.

How can you determine if a number is divisible by 3?

A number is divisible by 3 if the sum of its digits is divisible by 3. For 120, 1+2+0=3, which is divisible by 3, so 120 is divisible by 3.

What is the divisibility rule for 4?

A number is divisible by 4 if the number formed by its last two digits is divisible by 4. For 120, the last two digits are 20, which is divisible by 4.

Why is 5 a factor of 120?

5 is a factor of 120 because it ends with a 0 or a 5, which means it is divisible by 5.

What is the significance of the number 6 in relation to the factors of 120?

6 is a factor of 120 because it is divisible by both 2 and 3, and 120 is divisible by both of these numbers.

Why does the process of finding factors stop at 11 for the number 120?

The process stops at 11 because the square root of 120 is between 10 and 11, and factors come in pairs. Since 12 is already a factor, there are no factors greater than 11 that have not been accounted for.

How can you quickly determine the factors of 120 using the divisibility rules mentioned in the script?

You can quickly determine the factors of 120 by testing divisibility using the rules for 2 (even number), 3 (sum of digits divisible by 3), 4 (last two digits divisible by 4), 5 (ends in 0 or 5), and 6 (divisible by both 2 and 3). For other numbers, you can perform division or use the divisibility rules specific to those numbers.

What is the complete list of factors for the number 120?

The complete list of factors for 120 is 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120.

Why is 7 not a factor of 120?

7 is not a factor of 120 because when you divide 120 by 7, you get a remainder, indicating that 7 does not divide evenly into 120.

What is the significance of the number 8 as a factor of 120?

8 is a factor of 120 because when you divide 120 by 8, you get 15 with no remainder, indicating that 8 divides evenly into 120.

Outlines

00:00

🔢 Finding Factors of 120

This paragraph explains the process of finding all the factors of the number 120. It starts by identifying the smallest and largest factors, which are 1 and 120 respectively. The script then explores divisibility rules for 2, 3, and 4, demonstrating how to determine if 120 is divisible by these numbers. It shows that 120 is divisible by 2 because it's an even number, by 3 using the sum of its digits, and by 4 by examining the last two digits. The factors found are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120.

05:01

🔍 Continuing the Search for Factors

The second paragraph continues the process of identifying factors of 120. It begins with the divisibility rule for 5, noting that numbers ending in 0 or 5 are divisible by 5. The factors 5 and 24 are added to the list. The script then checks divisibility by 6, confirming that since 120 is divisible by both 2 and 3, it is also divisible by 6, resulting in factors 6 and 20. The divisibility by 7 is tested and found to be false, while 8 is confirmed as a factor with 15 as its pair. The divisibility by 9 is ruled out based on the sum of the digits not being divisible by 9. The factors 10 and 12 are added after confirming divisibility by 10. Lastly, the script tests and dismisses 11 as a factor, concluding the list of factors for 120.

Keywords

💡Factors

Factors, also known as divisors, are numbers that divide into another number without leaving a remainder. In the context of the video, the speaker is discussing the factors of 120, which are all the whole numbers that 120 is divisible by. The video uses the example of 1 and 120 being factors of 120, as every whole number is divisible by 1 and itself.

💡Divisibility

Divisibility refers to the property of a number being evenly divisible by another number. The video script explores divisibility rules for various numbers, such as 2, 3, 4, and 5, to determine if they are factors of 120. For instance, the speaker explains that 120 is divisible by 2 because it is an even number.

💡Even Number

An even number is any integer that is exactly divisible by 2. In the video, the concept is used to explain why 2 is a factor of 120, as 120 ends with a 0, indicating it is even.

💡Divisibility Rule

A divisibility rule is a method to determine whether a number is divisible by another number. The video mentions several rules, such as the rule for 3 (the sum of the digits must be divisible by 3) and for 4 (the last two digits must be divisible by 4). These rules are used to identify factors of 120.

💡Long Division

Long division is a method of dividing one number by another to find the quotient. The video script refers to long division as a way to verify the divisibility of 120 by numbers like 2, 3, and 4, and to find the corresponding factors.

💡Quotient

The quotient is the result of a division operation. In the video, the speaker calculates the quotient when dividing 120 by various numbers to find its factors, such as 120 divided by 2 equals 60, making 60 a factor of 120.

💡Remainder

A remainder is what is left over after division. The video script uses the concept of remainders to explain why certain numbers are not factors of 120, such as when dividing 120 by 7, a remainder indicates that 7 is not a factor.

💡Multiple

A multiple of a number is the product of that number and an integer. The video discusses finding multiples of numbers to identify factors of 120, such as stating that 120 is a multiple of 1, 2, 3, and so on.

💡Prime Factorization

Prime factorization is the process of breaking down a number into its prime factors. Although not explicitly mentioned, the video's process of finding factors is akin to prime factorization, as it identifies the prime numbers that multiply to give 120.

💡Descending Order

Descending order is a sequence from the highest to the lowest value. The video uses descending order to list the factors of 120, starting from the largest factor, 120 itself, and ending with the smallest factor, 1.

💡Tens Place

The tens place is the position of the second digit to the left in a number, representing the number of tens. The video uses the tens place to apply the divisibility rule for 4, stating that if the last two digits of a number are divisible by 4, then the whole number is divisible by 4.

Highlights

120 is divisible by 1 and itself, 120.

120 is divisible by 2 because it is an even number.

120 is divisible by 3 if the sum of its digits is divisible by 3.

120 is divisible by 4 if the last two digits form a number that is divisible by 4.

120 is divisible by 5 if it ends with a 0 or 5.

120 is divisible by 6 as it is divisible by both 2 and 3.

120 is divisible by 8, as shown by the division process.

120 is divisible by 10 because it ends with a 0.

120 is not divisible by 7 as shown by the division process.

120 is not divisible by 9 as the sum of its digits is not divisible by 9.

120 is divisible by 11, as shown by the division process.

Factors of 120 include 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120.

The largest factor of 120 is 120 itself.

The smallest factor of 120 is 1.

Divisibility rule for 2 is that the number must be even.

Divisibility rule for 3 is the sum of the digits must be divisible by 3.

Divisibility rule for 4 is the last two digits must form a number divisible by 4.

Divisibility rule for 5 is the number must end with 0 or 5.

Transcripts

Find all of the factors of 120.

Or another way to think about it, find all of the whole

numbers that 120 is divisible by.

So the first one, that's maybe obvious.

All whole numbers are divisible by 1.

So we could write 120 is equal to is to 1 times 120.

So let's write a factors list over here.

So this is going to be our factors list over here.

So we just found two factors.

We said, well, is it divisible by 1?

Well, every whole number is divisible by 1.

This is a whole number, so 1 is a factor at the low end.

1 is a factor.

That's its actual smallest factor, and its

largest factor is 120.

You can't have something larger than 120 dividing

evenly into 120.

121 will not go into 120.

So the largest factor on our factors list

is going to be 120.

Now let's think about others.

Let's think about whether is 2 divisible into 120?

So there's 120 equals 2 times something?

Well, when you look here, maybe you immediately

recognize that 120 is an even number.

It's ones place is a 0.

As as long as its ones place is a 0, 2, 4, 6 or 8, as long

as it's an even number, the whole number is even and the

whole number is divisible by 2.

And to figure out what you have to multiply by 2 to get

120, well, you can think of 120 as 12 times 10, or another

way to think about it, it's 2 times 6 times

10, or 2 times 60.

You could divide it out if you want.

You could say, OK, 2 goes into 120.

2 goes into 1 no times.

2 goes into 12 six times.

6 times 2 is 12.

Subtract.

You get 0.

Bring down the 0.

2 goes into 0 zero times.

0 times 2 is 0, and you get no remainder there, so it goes

sixty times.

So we have two more factors right here.

So we have the factors.

So we've established the next lowest one is 2, and the next

highest factor, if we're starting from the large end,

is going to be 60.

Now let's think about three.

Is 120 equal to 3 times something?

Well, we could just try to test and divide it from the

get go, but hopefully, you already know the

divisibility rule.

To figure out if something is divisible by 3, you add up its

digits, and if the sum is divisible

by 3, we're in business.

So if you take 120-- let me do it over here.

1 plus 2 plus 0, well, that's equal to 1 plus 2 is 3 plus 0

is 3, and 3 is definitely divisible by 3.

So 120 is going to be divisible by 3.

To figure what that number that you have to multiply by 3

is, you could do it in your head.

You could say, well, 3 goes into 12 four times, and then

you-- well, let me just do it out, just in case, just for

those of you who want to see it worked out.

3 goes into 12 four times.

4 times 3 is 12.

You subtract.

You're left with nothing here.

You bring down this 0.

3 goes into 0 zero times.

0 times 3 is 0.

Nothing left over.

So it goes into it forty times.

And the way to think of it in your head is this is the same

thing as 12 times 10.

12 divided by 3 is 4, but this is going to be 4 times 10,

because you have that 10 left over.

Whatever works for you.

Or you can just ignore the 0, divide by 3, you get a 4, and

then put the 0 back there.

Whatever works.

So we have two more factors.

At the low end, we have 3, and at the high end, we have a 40.

Now, let's see if 4 divisible into 120.

Now we saw the divisibility rule for 4 is you ignore

everything beyond the tens places and you just look at

the last two digits.

So if we're going to to think about whether 4 is divisible,

you just look at the last two digits.

The last two digits are 20.

20 is definitely divisible by 4, so 120 will be

divisible by 4.

4 is going to be a factor.

And to figure out what we have to multiply 4 by to get 120,

you could do it in your head.

You could say 12 divided by 4 is 3, so 120

divided by 4 is 30.

So we have two more factors: 4 and 30.

And you could work this out in long division if you want to

make sure that this works out, so let's keep going.

And then we have 120 is equal to-- is 5 a factor?

Is 5 times something equal to 120?

Well, you can't do that simple-- well, first of all,

we could just test is it divisible?

And 120 ends with a 0.

If you end with a 0 or a 5, you are divisible by 5.

So 5 definitely goes into it.

Let's figure out how many times.

So 5 goes into 120.

It doesn't go into 1.

It goes into 12 two times.

2 times 5 is 10.

Subtract.

You get 2.

Bring down the 0.

5 goes into 20 four times.

4 times 5 is 20, and then you subtract, and you have no left

over, as we expect, because it should go in evenly.

This number ends with a 0 or a 5.

Let me delete all of this so we can have our scratch space

to work with later on.

So 5 times 24 is also equal to 120, we have two more

factors: 5 and 24.

Let me clear up some space here because I think we're

going to be dealing with a lot of factors.

So let me move this right here.

Let me cut it and then let me paste it and move this over

here so we have more space for our factors.

So we have 5 and 24.

Let's move on to 6.

So 120 is equal to 6 times what?

Now, to be divisible by 6, you have to be

divisible by 2 and 3.

Now, we know that we're already divisible by 2 and 3,

so we're definitely going to be divisible by 6, and you

should hopefully be able to do this one in your head.

5 was a little bit harder to do in your head. but 120, you

could say, well, 12 divided by 6 is 2, and then you have that

0 there, so 120 divided by 6 would be 20.

And you could work it out in long division if you like.

So 6 times 20 are two more factors.

Now let's think about 7.

Let's think about 7 here.

7 is a very bizarre number, and just to test it, you could

think of other ways to do it.

Let's just try to divide 7 into 120.

7 doesn't go into 1.

It goes into 12 one time.

1 times 7 is 7.

You subtract.

12 minus 7 is 5.

Bring down the 0.

7 times 7 is 49, so it goes into it seven times.

7 times 7 is 49.

Subtract.

You have a remainder, so it does not divide evenly.

So 7 does not work.

Now let's think about 8.

Let's think about whether 8 works.

Let's think about 8.

I'll do the same process.

Let's take 8 into 120.

Let's just work it out.

And just as a little bit of a hint-- well, I'll

just work it out.

8 goes into 12-- it doesn't go into 1, so it

goes into 12 one time.

1 times 8 is 8.

Subtract there.

12 minus 8 is 4.

Bring down the 0.

8 goes into 40 five times.

5 times 8 is 40, and you're left with no remainder, so it

goes evenly.

So 120-- let me get rid of that.

120 is equal to 8 times 15, so let's add that to our factor

list. We now have an 8 and now we have a 15.

Now, is it divisible by 9?

Is 120 divisible by 9?

To test that out, you just add up the digits.

1 plus 2 plus 0 is equal to 3.

Well, that'll satisfy our 3 divisibility rule, but 3 is

not divisible by 9, so our number will not be

divisible by 9.

So 9 will not work out.

9 does not work out.

So let's move on to 10.

Well, this is pretty straightforward.

It ends in 0, so we will be divisible by 10.

So let me write that down.

120 is equal to 10 times-- and this is pretty

straightforward-- 10 times 12.

This is exactly what 120 is.

It's 10 times 12, so let's write those factors down.

10 and 12.

And then we have one number left.

We have 11.

We don't have to go above 11, because we already went

through 12, and we know that there aren't any factors above

that, because we were going in descending order, so we've

really filled in all the gaps.

You could try 11.

We could try it by hand, if you like.

11 goes into 120-- now you know, if with you know your

multiplication tables through 11, that this won't work, but

I'll just show you.

11 goes into 12 one time.

1 times 11 is 11.

Subtract.

1, bring down the 0.

11 goes into 10 zero times.

0 times 11 is 0.

you're left with a remainder of 10.

So 11 goes into 20 ten times with a remainder of 10.

It definitely does not go in evenly.

So we have all of our factors here: 1, 2, 3, 4, 5, 6, 8, 10,

12, 15, 20, 24, 30, 40, 60 and 120.

And we're done!