Fill The Blanks: LCM And GCD Math Puzzles

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Hey guys! Let's dive into some fun math puzzles involving the Least Common Multiple (LCM) and Greatest Common Divisor (GCD). These problems are all about finding the right numbers to fit specific conditions. It’s like detective work with numbers! So, grab your thinking caps, and let’s get started!

Understanding LCM and GCD

Before we jump into solving these puzzles, let's quickly recap what LCM and GCD mean.

  • Least Common Multiple (LCM): The LCM of two numbers is the smallest number that is a multiple of both. For example, the LCM of 4 and 6 is 12, because 12 is the smallest number that both 4 and 6 divide into evenly.
  • Greatest Common Divisor (GCD): The GCD of two numbers is the largest number that divides both numbers without leaving a remainder. For instance, the GCD of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18.

Knowing these definitions will help you approach and solve the puzzles more effectively. Remember, it's all about understanding the relationships between numbers!

Solving the Puzzles

Now, let's tackle those puzzles step by step. We'll break down each one to make it super clear and easy to follow. Get ready to fill in those blanks with the correct numbers!

a) LCM(25, ) = 75

In this problem, we need to find a number such that the Least Common Multiple (LCM) of 25 and that number is 75. Here’s how we can approach it:

  1. Factorize 25 and 75:

    • 25 = 5 * 5 = 52
    • 75 = 3 * 25 = 3 * 52

  2. Analyze the Factors:

    • The LCM (75) has a factor of 3, which is not present in 25. Therefore, the missing number must have 3 as a factor.
    • The highest power of 5 in 75 is 52, which is already present in 25. So, the missing number cannot have a higher power of 5 than 52.

  3. Find Possible Numbers:

    • The missing number must be a multiple of 3. Possible candidates include 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ..., 75.

  4. Check Each Candidate:

    • If we try 3, LCM(25, 3) = 75. So, 3 works!
    • If we try 15, LCM(25, 15) = 75. So, 15 also works!
    • If we try 75, LCM(25,75) = 75. So, 75 also works!

So, the possible numbers to fill in the blank are 3, 15, and 75.

b) GCD(, 60) = 12

Here, we're looking for a number where the Greatest Common Divisor (GCD) with 60 is 12. Let's break it down:

  1. Factorize 60 and 12:

    • 60 = 22 * 3 * 5
    • 12 = 22 * 3

  2. Analyze the Factors:

    • The GCD (12) has factors 22 and 3. This means the missing number must have these factors as well.
    • The missing number can have other factors, but it cannot have 5 as a factor (otherwise, the GCD would be larger than 12).

  3. Find Possible Numbers:

    • The missing number must be a multiple of 12. Possible candidates include 12, 24, 36, 48, 60, 72, 84, 96, ...

  4. Check Each Candidate:

    • If we try 12, GCD(12, 60) = 12. So, 12 works!
    • If we try 24, GCD(24, 60) = 12. So, 24 also works!
    • If we try 36, GCD(36, 60) = 12. So, 36 also works!
    • If we try 48, GCD(48, 60) = 12. So, 48 also works!
    • If we try 72, GCD(72, 60) = 12. So, 72 also works!
    • If we try 84, GCD(84, 60) = 12. So, 84 also works!
    • If we try 96, GCD(96, 60) = 12. So, 96 also works!

So, possible numbers are 12, 24, 36, 48, 72, 84, and 96.

c) LCM(, 28) = 140

Now, we need to find a number that, together with 28, has an LCM of 140. Let's break it down:

  1. Factorize 28 and 140:

    • 28 = 22 * 7
    • 140 = 22 * 5 * 7

  2. Analyze the Factors:

    • The LCM (140) has factors 22, 5, and 7. 28 already has 22 and 7, so the missing number must introduce the factor of 5.
    • The missing number must be a multiple of 5 but can have other factors as well.

  3. Find Possible Numbers:

    • The missing number must have 5 as a factor. Possible candidates include 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, ...

  4. Check Each Candidate:

    • If we try 5, LCM(5, 28) = 140. So, 5 works!
    • If we try 20, LCM(20, 28) = 140. So, 20 also works!
    • If we try 35, LCM(35, 28) = 140. So, 35 also works!
    • If we try 70, LCM(70, 28) = 140. So, 70 also works!
    • If we try 140, LCM(140, 28) = 140. So, 140 also works!

So, possible numbers are 5, 20, 35, 70 and 140.

d) GCD(90, ) = 18

In this case, we need to find a number that has a GCD of 18 with 90. Let's see how to do it:

  1. Factorize 90 and 18:

    • 90 = 2 * 32 * 5
    • 18 = 2 * 32

  2. Analyze the Factors:

    • The GCD (18) has factors 2 and 32. This means the missing number must have these factors but cannot have 5 as a factor (or the GCD would be higher).

  3. Find Possible Numbers:

    • The missing number must be a multiple of 18. Possible candidates include 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, ...

  4. Check Each Candidate:

    • If we try 18, GCD(90, 18) = 18. So, 18 works!
    • If we try 36, GCD(90, 36) = 18. So, 36 also works!
    • If we try 54, GCD(90, 54) = 18. So, 54 also works!
    • If we try 72, GCD(90, 72) = 18. So, 72 also works!
    • If we try 108, GCD(90, 108) = 18. So, 108 also works!
    • If we try 126, GCD(90, 126) = 18. So, 126 also works!
    • If we try 144, GCD(90, 144) = 18. So, 144 also works!

So, possible numbers are 18, 36, 54, 72, 108, 126, and 144.

e) LCM(, 42) = 210

For this one, we need to find a number such that the LCM with 42 is 210. Here’s the breakdown:

  1. Factorize 42 and 210:

    • 42 = 2 * 3 * 7
    • 210 = 2 * 3 * 5 * 7

  2. Analyze the Factors:

    • The LCM (210) has factors 2, 3, 5, and 7. 42 already has 2, 3, and 7, so the missing number must introduce the factor of 5.

  3. Find Possible Numbers:

    • The missing number must be a multiple of 5. Possible candidates include 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, ..., 210.

  4. Check Each Candidate:

    • If we try 5, LCM(5, 42) = 210. So, 5 works!
    • If we try 10, LCM(10, 42) = 210. So, 10 also works!
    • If we try 15, LCM(15, 42) = 210. So, 15 also works!
    • If we try 30, LCM(30, 42) = 210. So, 30 also works!
    • If we try 35, LCM(35, 42) = 210. So, 35 also works!
    • If we try 70, LCM(70, 42) = 210. So, 70 also works!
    • If we try 105, LCM(105, 42) = 210. So, 105 also works!
    • If we try 210, LCM(210, 42) = 210. So, 210 also works!

So, possible numbers are 5, 10, 15, 30, 35, 70, 105 and 210.

f) GCD(, 120) = 15

Lastly, we need to find a number that has a GCD of 15 with 120. Let’s break it down:

  1. Factorize 120 and 15:

    • 120 = 23 * 3 * 5
    • 15 = 3 * 5

  2. Analyze the Factors:

    • The GCD (15) has factors 3 and 5. This means the missing number must have these factors but cannot have 2 as a factor (or the GCD would be higher).

  3. Find Possible Numbers:

    • The missing number must be a multiple of 15. Possible candidates include 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, 195, ...

  4. Check Each Candidate:

    • If we try 15, GCD(15, 120) = 15. So, 15 works!
    • If we try 45, GCD(45, 120) = 15. So, 45 also works!
    • If we try 75, GCD(75, 120) = 15. So, 75 also works!
    • If we try 105, GCD(105, 120) = 15. So, 105 also works!
    • If we try 135, GCD(135, 120) = 15. So, 135 also works!
    • If we try 165, GCD(165, 120) = 15. So, 165 also works!
    • If we try 195, GCD(195, 120) = 15. So, 195 also works!

So, possible numbers are 15, 45, 75, 105, 135, 165, and 195.

Conclusion

Finding the missing numbers for LCM and GCD problems can be a fun and engaging way to sharpen your math skills. Remember, it’s all about understanding the factors and multiples involved. Keep practicing, and you’ll become a pro at solving these puzzles! Great job, guys! Now, go tackle more math challenges with confidence!

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