What Is The Greatest Common Factor Of 84 And 90
Hey there, math adventurer! Ever stared at two numbers, like 84 and 90, and wondered, "What's the biggest number that can evenly divide both of them?" It sounds like a riddle, right? Well, it's actually a super cool math concept called the Greatest Common Factor, or GCF for short. Think of it as the ultimate common ground between two numbers.
Let's break it down. We've got our dynamic duo: 84 and 90. We're on a mission to find their GCF. Don't worry, this isn't going to be one of those stuffy math lessons where you start to drift off thinking about what's for lunch. We're going to tackle this like a fun puzzle, one step at a time.
First off, what does it mean for a number to "evenly divide" another? It means when you divide the first number by the second, you get a whole number with no remainder. Like, 10 divided by 5 is 2, no leftover bits. But 10 divided by 3? That's 3 with a remainder of 1. So, 3 doesn't evenly divide 10. Easy peasy, lemon squeezy, right?
Now, for 84 and 90. We need to find a number that goes into 84 perfectly and goes into 90 perfectly. And not just any number, but the biggest one that can do both jobs. It's like finding the biggest piece of pizza that fits perfectly into two different-sized boxes. (Okay, maybe that analogy is a bit stretched, but you get the idea!)
There are a couple of fun ways to find the GCF. The first one is like going on a treasure hunt. We're going to list out all the factors of each number. What are factors? They are the numbers that multiply together to make our original number. So, for example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because: 1 x 12 = 12 2 x 6 = 12 3 x 4 = 12
See? Each of those numbers perfectly divides 12. And those are all the numbers that do. So, let's start our treasure hunt with 84. What numbers can we multiply to get 84?
We know that 1 is always a factor of any number. So, 1 x 84 = 84. That's a definite keeper.
What about 2? Yep, 84 is an even number, so 2 goes into it. 2 x 42 = 84. So, 2 and 42 are factors.
How about 3? To check if a number is divisible by 3, we can add up its digits. For 84, 8 + 4 = 12. And 12 is divisible by 3 (12 / 3 = 4), so 84 is also divisible by 3! Let's see... 84 divided by 3 is 28. So, 3 and 28 are factors.
Next up, 4. If a number is divisible by 4, the last two digits are divisible by 4. In 84, the last two digits are 84. Is 84 divisible by 4? Yes, 84 / 4 = 21. So, 4 and 21 are factors.
What about 5? Numbers divisible by 5 end in a 0 or a 5. 84 doesn't. So, 5 is out.
Let's try 6. If a number is divisible by both 2 and 3, it's divisible by 6. We already know 84 is divisible by both 2 and 3. So, it's divisible by 6! 84 divided by 6 is... drumroll please... 14! So, 6 and 14 are factors.
How about 7? This one you might just know or have to try out. Does 7 go into 84? Let's see... 7 x 10 = 70. We need 14 more. And 7 x 2 = 14. So, 7 x (10 + 2) = 7 x 12 = 84. Yep, 7 and 12 are factors.
Okay, what's next? 8? Let's try. 84 divided by 8. Hmm, 8 x 10 = 80. That leaves 4. Nope, 8 doesn't go into 84 evenly.
What about 9? To check divisibility by 9, we add the digits again. 8 + 4 = 12. Is 12 divisible by 9? Nope. So, 9 is out.
Next is 10. Nope, doesn't end in 0.
What about 11? This is a bit trickier. For 84, alternating sum of digits: 8 - 4 = 4. Not 0 or a multiple of 11. So, 11 is out.
And then we have 12! We already found that 7 x 12 = 84. So, 12 is a factor.
Now, notice something cool. When we found that 7 x 12 = 84, we had already found 12 as a potential factor. And since 12 is greater than 7, we can stop looking for new pairs from this side. We've essentially met in the middle! Our factors for 84 are:
1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
Phew! That was a bit of a climb, but we got there. Now, let's do the same for our other number, 90. Ready for another treasure hunt?
Again, 1 is always a factor. 1 x 90 = 90.
90 is even, so 2 is a factor. 2 x 45 = 90.
For 3, let's add the digits: 9 + 0 = 9. And 9 is divisible by 3 (9 / 3 = 3). So, 90 is divisible by 3! 90 divided by 3 is 30. So, 3 and 30 are factors.
4? The last two digits are 90. Is 90 divisible by 4? 4 x 20 = 80, leaves 10. Nope. So, 4 is out.
5? Yes! It ends in a 0. 5 x 18 = 90. So, 5 and 18 are factors.
6? Is it divisible by both 2 and 3? Yes! So, 90 is divisible by 6. 90 divided by 6 is 15. So, 6 and 15 are factors.
7? Let's try. 7 x 10 = 70. Needs 20 more. 7 doesn't go into 20. So, 7 is out.
8? 8 x 10 = 80. Leaves 10. Nope. So, 8 is out.
9? We know 9 + 0 = 9, which is divisible by 9. So, 90 is divisible by 9! 90 divided by 9 is 10. So, 9 and 10 are factors.
And there we have it! We found 9 and 10. Since 10 is larger than 9, we can stop here. The factors of 90 are:
1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
Okay, deep breaths! We've got our lists of factors for both 84 and 90.
Now, for the grand finale of this treasure hunt, we need to find the common factors. These are the numbers that appear on both lists. Let's compare them:
Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
See them? The common factors are: 1, 2, 3, and 6. These are the numbers that can divide both 84 and 90 perfectly. They are the mutual friends of 84 and 90!
But the question asks for the Greatest Common Factor. Which one of those common factors is the biggest? Looking at our list (1, 2, 3, 6), the biggest one is clearly 6!
So, drumroll please... The Greatest Common Factor of 84 and 90 is 6!
Ta-da! We did it! We navigated the land of factors and emerged victorious with the GCF.
Now, there's another super-duper handy way to find the GCF, especially for bigger numbers. It's called the Prime Factorization Method. Don't let the fancy name scare you. It just means we're going to break down each number into its "prime" building blocks.
What are prime numbers? They are numbers greater than 1 that can only be divided evenly by 1 and themselves. Think of them as the indivisible atoms of the number world. Examples are 2, 3, 5, 7, 11, 13, and so on. Numbers like 4, 6, 8, 9, 10? Not prime. They can be broken down further.
Let's break down 84 into its prime factors. We can do this using a factor tree. It looks like a little upside-down tree:
Start with 84 at the top. Branch out into any two factors. Let's use 2 and 42 (since 2 x 42 = 84).
84 / \ 2 42
Is 2 prime? Yes! So, we circle it. Now, let's break down 42. We know 2 x 21 = 42.
84 / \ 2 42 / \ 2 21
2 is prime. Now, break down 21. We know 3 x 7 = 21.
84 / \ 2 42 / \ 2 21 / \ 3 7
Are 3 and 7 prime? Yes! So, we circle them too. We've reached the end of our branches when all the numbers at the end are prime.
So, the prime factorization of 84 is 2 x 2 x 3 x 7.
Now, let's do the same for 90. Start with 90.
Let's branch into 9 and 10 (since 9 x 10 = 90).
90 / \ 9 10
Is 9 prime? Nope! 3 x 3 = 9. Is 10 prime? Nope! 2 x 5 = 10.
90 / \ 9 10 / \ / \ 3 3 2 5
Are 3, 3, 2, and 5 prime? Yes, they are! So, we circle them all.
The prime factorization of 90 is 2 x 3 x 5.
Now, to find the GCF using prime factors, we look for the common prime factors and multiply them together. It's like finding the ingredients that both recipes share!
Prime factors of 84: 2, 2, 3, 7
Prime factors of 90: 2, 3, 5
What prime factors do they both have? They both have one 2, and they both have one 3.
So, we multiply those common prime factors: 2 x 3 = 6.
And there it is again! The Greatest Common Factor of 84 and 90 is 6. This method can be a lifesaver when the numbers get much bigger.
Isn't it amazing how these numbers have these hidden connections? Finding the GCF isn't just about numbers; it's about finding common ground, about shared elements. It’s a little bit of math magic that helps simplify things and understand the relationships between numbers.
So, the next time you see two numbers, remember the GCF. It’s the biggest number that can play nicely with both of them. And that, my friend, is a pretty neat thing to know. Keep exploring, keep questioning, and remember that even the most complex-looking math problems can be fun puzzles waiting to be solved. You've got this!