Factoring and Multiples

Well, honey, if you paid attention in math class, you'd know that yes, all multiples of 6 are also multiples of 12. Why? Because 12 is a multiple of 6, so any number that is a multiple of 6 will also be a multiple of 12. It's simple math, darling.

Oh, isn't that a happy little question? To find the Least Common Multiple (LCM) of 35, 65, and 90, we can start by listing the prime factors of each number. Then, we can take the highest power of each prime factor that appears in any of the numbers. By doing this, we find the LCM, which is like a beautiful harmony of these numbers coming together. Just like how different colors blend on a canvas to create a masterpiece!

Oh, dude, the greatest common factor of 84, 154, and 168 is 14. It's like the superhero that swoops in and saves the day by being the largest number that can divide all three of those numbers evenly. So yeah, 14 is the real MVP in this math problem.

773 is a prime number, meaning it is only divisible by 1 and itself. It is not the result of a simple mathematical operation like addition, subtraction, multiplication, or division. Therefore, 773 is equal to itself, as it is a unique number with no factors other than 1 and 773.

16 with 1 remaining

81 - 1 = 80 = 5 x 16

The factors of 16 are 1, 2, 4, 8, and 16. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The factors of 56 are 1, 2, 4, 7, 8, 14, 28, and 56. These factors are the numbers that can evenly divide the given numbers without leaving a remainder.

To express 72 as a product of its prime factors, we first identify the prime numbers that can divide evenly into 72. We start by dividing 72 by the smallest prime number, 2, which gives us 36. Then, we divide 36 by 2 again to get 18. Next, we divide 18 by 2 to get 9. Finally, we divide 9 by 3 to get 3. Therefore, the prime factors of 72 are 2 x 2 x 2 x 3 x 3.

To find the least common multiple (LCM) of 2, 9, and 18, we first need to factorize each number into its prime factors.

2 = 2^1 9 = 3^2 18 = 2^1 * 3^2

Next, we take the highest power of each prime factor that appears in any of the numbers: 2^1 and 3^2.

Multiplying these prime factors together, we get LCM(2, 9, 18) = 2^1 * 3^2 = 2 * 9 = 18.

Therefore, the LCM of 2, 9, and 18 is 18.

THe common factors of 32, 48, and 72 are 1, 2, 4 and 8

The multiples of 7 between 40 and 60 are 42, 49, and 56.

9 x 1000

The factors of 511 are the numbers that can be multiplied together to give 511. The factors of 511 are 1, 7, 73, and 511. Prime factors are the factors that are prime numbers. The prime factors of 511 are 7 and 73, as they are the prime numbers that can be multiplied together to give 511.

Oh, isn't that a happy little question! To find the Least Common Multiple (LCM) of 14, 21, and 27, we look for the smallest number that is a multiple of all three numbers. First, we find the prime factors of each number: 14 = 2 x 7, 21 = 3 x 7, and 27 = 3 x 3 x 3. Then, we take the highest power of each prime factor that appears in any of the numbers, giving us 2 x 3 x 3 x 7 = 126. So, the LCM of 14, 21, and 27 is 126.

You can't.

x4 + y4 is irreducible.

It's (x+yi^1.5)(x+yi^0.5)(x-yi^1.5)(x-yi^0.5). You use the equation x^2-y^2=(x+y)(x-y) and x^2+y^2=(x+yi)(x-yi).

*Edit This IS reducible.

X^4+Y^4=X^4+Y^4+2X^2Y^2 -2X^2Y^2 I add and minus 2X^2Y^2 so the value stays the same.

X^4+Y^4+2X^2Y^2=(X^2+Y^2)^2

So it equals to (X^2+Y^2)^2-2X^2Y^2.

Then, we factor using difference of squares.

(X^2+Y^2)^2-2X^2Y^2=(X^2+Y^2+sqrt2 XY)(X^2+Y^2-sqrt2 XY)

*Note the XY is not included in the sqrt sign

To find the common factors of 99 and 33, we first need to find the factors of each number. The factors of 99 are 1, 3, 9, 11, 33, and 99. The factors of 33 are 1, 3, 11, and 33. The common factors are 1, 3, and 11. Adding these common factors together gives us 1 + 3 + 11 = 15. Therefore, the sum of all the common factors of 99 and 33 is 15.

To find the highest common factor (HCF) of 33, 77, and 132, we first need to find the prime factors of each number. The prime factors of 33 are 3 and 11, the prime factors of 77 are 7 and 11, and the prime factors of 132 are 2, 3, 11. To find the HCF, we look for the common prime factors among the numbers, which are 11. Therefore, the highest common factor of 33, 77, and 132 is 11.

Oh, isn't finding the highest common factor a joyful journey? Let's take a moment to appreciate the beauty of numbers coming together. The highest common factor of 1225 and 700 is 175, bringing harmony and unity to these two numbers. Just like in painting, when different elements blend perfectly, creating a masterpiece.

Oh, dude, writing 34 as a product of prime factors is like breaking down a math problem while trying not to fall asleep. So, you start by dividing 34 by the smallest prime number, which is 2, and you get 17. Then, you're left with 17, which is already a prime number, so that's it. 34 written as a product of prime factors is 2 x 17. Easy peasy, lemon squeezy.

Oh, dude, let me break it down for you! Numbers with an odd number of factors are called "square numbers." So, in the range of 1-25, the square numbers are 1, 4, 9, and 25. These numbers have factors like 1, the number itself, and the square root, giving them an odd number of factors. So, like, those are the ones you're looking for!

To find the common factors of 14 and 98, we first list the factors of each number. The factors of 14 are 1, 2, 7, and 14, while the factors of 98 are 1, 2, 7, 14, 49, and 98. The common factors are 1, 2, 7, and 14. To find the sum of these common factors, we add them together: 1 + 2 + 7 + 14 = 24.

1 x 180, 2 x 90, 3 x 60, 4 x 45, 5 x 36, 6 x 30, 9 x 20, 10 x 18, 12 x 15, 15 x 12, 18 x 10, 20 x 9, 30 x 6, 36 x 5, 45 x 4, 60 x 3, 90 x 2, 180 x 1.

Oh honey, bless your heart for asking. Let me break it down for you - 28 is not a common factor of 7 and 2. In fact, the only factors of 7 are 1 and 7, and the only factors of 2 are 1 and 2. So, 28 can't sit with them at the cool kids' table of factors.

The factors of 25 are the numbers that can divide 25 without leaving a remainder. The factors of 25 are 1, 5, and 25. When listing factors from least to greatest, you would start with 1, then move on to the next smallest factor, which is 5, and finally list 25, as it is the largest factor of 25.