Factoring and Multiples

Alright, sweetheart, let's keep it simple for you. Every even number has 2 as a factor. So, grab a calculator if you need to, and start counting by twos until you get tired. Just don't come crying to me when you realize there's an infinite number of them.

Oh, dude, the positive difference between the number of distinct prime factors of 15 squared (225) and 64 cubed (262,144) is like 4. Prime factors are like those VIP numbers that can't be divided by anything except 1 and themselves. So, 225 has 2 distinct prime factors (3 and 5), and 262,144 has 1 distinct prime factor (2). Subtract 1 from 2, and you get 1. Wow, math can be fun, right?

Numbers that are divisible by 1024 are multiples of 1024, meaning they can be expressed as 1024 times an integer. In other words, any number that can be written as 1024 x n, where n is an integer, is divisible by 1024. Some examples of numbers divisible by 1024 are 1024, 2048, 3072, and so on.

The prime factors of 180 are the prime numbers that can be multiplied together to equal 180. To find the prime factors of 180, we start by dividing 180 by the smallest prime number, which is 2. Continuing this process, we find that the prime factors of 180 are 2, 2, 3, 3, and 5, represented as 2^2 * 3^2 * 5.

Oh, dude, multiples of 4 are like 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, and so on. So, if you're looking for multiples of 4 that are not 48, just ignore 48 on that list. It's like a party where everyone's invited except for 48.

Well, honey, the common multiples of 6, 8, and 12 are numbers that can be divided by all three without leaving a pesky remainder. So, grab your calculator and start multiplying those numbers until you find the ones that satisfy all three divas. Good luck, darling!

21

The highest common factor (HCF) of 22 and 58 is the largest positive integer that divides both numbers without leaving a remainder. To find the HCF, you can use methods like prime factorization or the Euclidean algorithm. In this case, the prime factorization of 22 is 2 x 11 and the prime factorization of 58 is 2 x 29. The common factor between them is 2, so the HCF of 22 and 58 is 2.

Oh, dude, the Highest Common Factor (HCF) of 490 and 1050 is 70. It's like the cool kid that hangs out with both 490 and 1050 without any drama. So, yeah, 70 is the number that divides both 490 and 1050 without leaving any remainder.

The least common multiple (LCM) is often also called the lowest common multiple or smallest common multiple. Keep in mind that these different terms all refer to the same thing: the smallest positive integer which is a multiple of two or more numbers.

The least common multiple of 270 and 360 is 1080.

The least common multiple (LCM) of 6a and 8a is the smallest number that is divisible by both 6a and 8a. To find the LCM, we need to first find the prime factors of each number. The prime factors of 6a are 2 x 3 x a, and the prime factors of 8a are 2 x 2 x 2 x a. The LCM is the product of all the unique prime factors with the highest power, so the LCM of 6a and 8a is 2 x 2 x 2 x 3 x a, which simplifies to 24a.

Oh, dude, the LCM of 32 and 15 is like the smallest number that both 32 and 15 can divide evenly into. So, you take the prime factors of each number, which are 2^5 and 3 * 5, then you just multiply the highest power of each prime factor together. That gives you 2^5 * 3 * 5, which equals 480. So, the LCM of 32 and 15 is 480.

To find the Greatest Common Factor (GCF) of 35, 65, and 90, we need to first find the prime factorization of each number.

35 = 5 x 7 65 = 5 x 13 90 = 2 x 3 x 3 x 5

Next, we identify the common prime factors among the numbers, which are 5. Therefore, the GCF of 35, 65, and 90 is 5.

The number eight is divisible by 1, 2, 4, and 8. This is because 8 divided by 1 equals 8, 8 divided by 2 equals 4, 8 divided by 4 equals 2, and 8 divided by 8 equals 1. These are the numbers that evenly divide into eight without leaving a remainder.

The prime factorization of 25n^2 is 5 * 5 * n * n, or simply 5^2 * n^2. This means that the prime factors of 25n^2 are 5 and n, each raised to the power of 2. In other words, 25n^2 can be expressed as the product of 5 squared and n squared.

When something increases by a factor of 16, it means that the quantity has been multiplied by 16. For example, if you have a value of 5 and it increases by a factor of 16, the new value would be 5 x 16 = 80. This is a significant increase compared to just adding 16 to the original value.

Oh, what a happy little question! Let's see here... multiples of 4 less than 100 are numbers that you get when you multiply 4 by another number and the result is less than 100. So, we have 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, and 96. Just like painting, finding multiples can be a fun and calming experience.

Oh, isn't that just a happy little question! Let's take a moment to appreciate the factors of 15 and 65. The common factors of 15 and 65 are 1 and 5, as they can both divide evenly into both numbers. It's like a little math puzzle waiting to be solved, and you're doing great just by asking!

To find 1/8 of 88, you would divide 88 by 8. This calculation results in 11, so 1/8 of 88 is 11. This can be verified by multiplying 11 by 8, which equals 88.

Oh, isn't that a lovely question! To find the highest common factor (HCF) of 9, 54, and 18, we first look at the factors of each number. The factors of 9 are 1, 3, and 9; the factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54; and the factors of 18 are 1, 2, 3, 6, 9, and 18. The largest number that appears in the factors of all three numbers is 9, so the HCF of 9, 54, and 18 is 9. Isn't that just a happy little number to find?

To find the Greatest Common Factor (GCF) of 45 and 65, we first need to list the factors of each number. The factors of 45 are 1, 3, 5, 9, 15, and 45, while the factors of 65 are 1, 5, 13, and 65. The largest number that appears in both lists is 5, so the GCF of 45 and 65 is 5.

The highest common factor of 875 is 125. To find the highest common factor, you need to identify all the factors of the number and then determine the largest factor that is common to all of them. In this case, the factors of 875 are 1, 5, 25, 35, 125, and 875. The highest common factor among these factors is 125.

The term common factors specifically refers to the factors shared by two or more numbers. Since 144 is a single number, you could say that it shares all of its factors with itself.
The factors of 144 are: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, and 144.