GCD Problems: Solving Math Questions Step-by-Step

Hey guys! Today, we're diving into some cool math problems all about the Greatest Common Divisor (GCD), also known as Ebob in some places. We'll break down each question step by step, so it’s super easy to follow. Let's get started!

1. What is the Greatest Common Divisor (GCD) of 30 and 35?

So, what exactly is the Greatest Common Divisor (GCD)? Simply put, it's the largest number that can divide two or more numbers without leaving a remainder. To find the GCD of 30 and 35, we need to list their factors and identify the largest one they have in common.

First, let's list the factors of 30:

1, 2, 3, 5, 6, 10, 15, 30

Now, let's list the factors of 35:

1, 5, 7, 35

Comparing the two lists, we can see that the common factors are 1 and 5. The largest among these is 5. Therefore, the GCD of 30 and 35 is 5.

Why is GCD important? Understanding GCD helps simplify fractions, solve problems related to division and ratios, and even in more advanced topics like cryptography. Knowing how to find the GCD manually gives you a solid foundation for tackling more complex math challenges. For example, when you're trying to reduce a fraction to its simplest form, finding the GCD of the numerator and denominator is crucial. Also, in real-life scenarios, like dividing items into equal groups, GCD comes in handy. Imagine you have 30 apples and 35 oranges, and you want to make identical fruit baskets. The largest number of baskets you can make where each basket has the same number of apples and oranges is 5. Each basket would have 6 apples and 7 oranges. Pretty neat, right?

2. What is the Greatest Common Divisor (GCD) of 150 and 180?

Alright, let's tackle another GCD problem. This time, we want to find the GCD of 150 and 180. We'll follow the same method as before: list the factors of each number and find the largest one they share.

Factors of 150:

1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150

Factors of 180:

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

By comparing the two lists, we identify the common factors: 1, 2, 3, 5, 6, 10, 15, and 30. The largest of these common factors is 30. Therefore, the GCD of 150 and 180 is 30.

Alternative Method: Prime Factorization

Another cool way to find the GCD is by using prime factorization. First, break down each number into its prime factors:

• 150 = 2 x 3 x 5 x 5 = 2 x 3 x 5²
• 180 = 2 x 2 x 3 x 3 x 5 = 2² x 3² x 5

Now, identify the common prime factors and their lowest powers: 2¹, 3¹, and 5¹. Multiply these together:

2 x 3 x 5 = 30

So, using prime factorization, we also find that the GCD of 150 and 180 is 30. This method is particularly useful when dealing with larger numbers.

3. Find the Largest Number That Leaves a Remainder of 5 When Dividing Both 96 and 57.

This problem is a bit different, but still relies on the concept of GCD. We're looking for a number that, when it divides 96 and 57, leaves a remainder of 5 in both cases. This means if we subtract 5 from both numbers, the result should be perfectly divisible by the number we're looking for.

Let's subtract 5 from 96 and 57:

• 96 - 5 = 91
• 57 - 5 = 52

Now, we need to find the GCD of 91 and 52. Let's list their factors:

Factors of 91:

1, 7, 13, 91

Factors of 52:

1, 2, 4, 13, 26, 52

The common factors are 1 and 13. The largest common factor is 13. Therefore, the largest number that leaves a remainder of 5 when dividing both 96 and 57 is 13.

Checking Our Answer

Let's verify this. When we divide 96 by 13:

96 ÷ 13 = 7 remainder 5

And when we divide 57 by 13:

57 ÷ 13 = 4 remainder 5

So, our answer checks out! This type of problem often appears in number theory and requires a bit of logical thinking to set up correctly.

4. A Garden Has Sides of Length 60 m and 72 m. Trees Will Be Planted Around the Garden at Equal Intervals, with a Tree at Each Corner. What is the Largest Possible Interval Between the Trees?

This is a classic problem that combines geometry with GCD. We need to find the largest possible interval between trees planted around a rectangular garden, with a tree at each corner. This interval must be a common divisor of the lengths of the sides of the garden.

So, we need to find the GCD of 60 and 72. Let's list the factors of each number:

Factors of 60:

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Factors of 72:

1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

By comparing the two lists, we find the common factors: 1, 2, 3, 4, 6, and 12. The largest of these common factors is 12. Therefore, the largest possible interval between the trees is 12 meters.

Visualizing the Solution

Imagine the garden. Along the 60-meter side, you would have trees planted every 12 meters, resulting in 60/12 = 5 sections. Since there’s a tree at both ends, you'll have 6 trees along that side. Along the 72-meter side, you'd have trees planted every 12 meters, resulting in 72/12 = 6 sections, and thus 7 trees along that side. This ensures that the trees are evenly spaced and there’s a tree at each corner. Problems like this are commonly found in landscaping and design, where equal spacing and efficient use of resources are important.

Conclusion

So, there you have it! We've tackled four different types of GCD problems, from finding the GCD of two numbers to more complex scenarios involving remainders and geometric arrangements. Understanding GCD is super useful in math and real-life situations. Keep practicing, and you'll become a GCD pro in no time! Keep an eye out for more math tips and tricks. Happy problem-solving, guys!