Prime Factorization, GCD, LCM, And Simplification

Nov 3, 2025 by Admin 50 views

Hey guys! Today, let's break down some numbers and explore prime factorization, greatest common divisors (GCD), least common multiples (LCM), and simplification. We'll start with a = 2520 and b = 1050. Let's dive in!

1. Prime Factorization of a and b

First off, let's find the prime factors of a = 2520 and b = 1050. Prime factorization is all about breaking down a number into its prime number building blocks. It's like reverse engineering a number to see what prime numbers multiply together to give you the original number. It's super useful in many areas of math, including finding the GCD and LCM, which we'll do next! For the number 2520, we begin by dividing it by the smallest prime number, which is 2. We keep dividing by 2 until it's no longer possible. Then, we move on to the next prime number, 3, and repeat the process. We continue this until we are left with only prime factors. Let's kick things off with a = 2520. By repeatedly dividing by prime numbers, we find that 2520 can be expressed as a product of its prime factors. So we get:

2520 = 2 × 1260
1260 = 2 × 630
630 = 2 × 315
315 = 3 × 105
105 = 3 × 35
35 = 5 × 7

Putting it all together, 2520 = 2³ × 3² × 5 × 7. Now, let's move on to b = 1050. We'll do the same thing here, breaking it down into its prime factors step by step. Start by dividing by the smallest prime number, 2, and continue with the next prime numbers until we are left with only prime factors. Here’s the breakdown:

1050 = 2 × 525
525 = 3 × 175
175 = 5 × 35
35 = 5 × 7

So, 1050 = 2 × 3 × 5² × 7. And there you have it! Both numbers are now expressed in terms of their prime factors. This is the foundation for finding the GCD and LCM.

2. Finding GCD(a, b) and LCM(a, b)

Now that we have the prime factorizations of a and b, finding the Greatest Common Divisor (GCD) and the Least Common Multiple (LCM) becomes much easier. The GCD is the largest number that divides both a and b without leaving a remainder. The LCM, on the other hand, is the smallest number that both a and b divide into without leaving a remainder. To find the GCD, we look at the prime factors that a and b have in common. For each common prime factor, we take the lowest power that appears in either factorization. Then, we multiply these together to get the GCD. Looking at the prime factorizations:

a = 2³ × 3² × 5 × 7
b = 2 × 3 × 5² × 7

The common prime factors are 2, 3, 5, and 7. The lowest powers of these factors are:

2¹ (from b)
3¹ (from b)
5¹ (from a)
7¹ (from both)

So, GCD(2520, 1050) = 2¹ × 3¹ × 5¹ × 7¹ = 2 × 3 × 5 × 7 = 210. Now, let's find the LCM. To find the LCM, we look at all the prime factors that appear in either a or b. For each prime factor, we take the highest power that appears in either factorization. Then, we multiply these together to get the LCM. Again, looking at the prime factorizations:

a = 2³ × 3² × 5 × 7
b = 2 × 3 × 5² × 7

The highest powers of these factors are:

2³ (from a)
3² (from a)
5² (from b)
7¹ (from both)

So, LCM(2520, 1050) = 2³ × 3² × 5² × 7¹ = 8 × 9 × 25 × 7 = 12600. Therefore, we have GCD(a, b) = 210 and LCM(a, b) = 12600. Knowing the GCD and LCM is super useful in simplifying fractions and solving various mathematical problems!

3. Simplifying the Fraction a/b and Simplifying √(ab)

Let's simplify the fraction a/b and simplify the number √(ab). Simplifying fractions and square roots is a common task in math, and it's all about making things as neat and tidy as possible. It not only makes the numbers easier to work with but also helps in understanding the relationships between numbers. Let's tackle a/b first. We have a = 2520 and b = 1050, so the fraction is 2520/1050. To reduce this fraction to its simplest form, we need to divide both the numerator and the denominator by their greatest common divisor (GCD), which we found earlier to be 210. So, we divide both 2520 and 1050 by 210:

2520 ÷ 210 = 12
1050 ÷ 210 = 5

Therefore, the simplified fraction is 12/5. Now, let's simplify √(ab). We have a = 2520 and b = 1050, so we need to find the square root of their product. We already know the prime factorizations of a and b:

a = 2³ × 3² × 5 × 7
b = 2 × 3 × 5² × 7

So, ab = (2³ × 3² × 5 × 7) × (2 × 3 × 5² × 7) = 2⁴ × 3³ × 5³ × 7². Now, to find the square root, we take half the power of each prime factor:

√(ab) = √(2⁴ × 3³ × 5³ × 7²) = 2² × 3^(3/2) × 5^(3/2) × 7 = 2² × 3 × 5 × 7 × √(3 × 5) = 4 × 3 × 5 × 7 × √15 = 420√15.

So, √(ab) simplifies to 420√15. We've now simplified both the fraction a/b and the number √(ab), making them much easier to understand and work with!

4. Determining the Smallest Number

Unfortunately, the prompt ends abruptly at “Determine the smallest number.” To provide a complete answer, we need more context. What number are we trying to find? What conditions or properties must it satisfy? Without additional information, it's impossible to determine the smallest number. To help me give you a complete answer, please provide the full problem statement or clarify the context in which you're looking for the smallest number. For instance, are we looking for the smallest number that, when multiplied by a or b, results in a perfect square? Or the smallest number that can be divided by both a and b without any remainder? Once I have this information, I can guide you through the steps to find the smallest number that meets your criteria. Add more details and I’ll be happy to help!

In Summary

We've successfully completed the following steps:

Decomposed a = 2520 and b = 1050 into their prime factors.
Calculated the GCD(a, b) = 210 and LCM(a, b) = 12600.
Simplified the fraction a/b to 12/5.
Simplified √(ab) to 420√15.

Remember, math is all about practice. Keep working at it, and you'll get better with each problem you solve!