Factors Of 50: Find Odd And Even Factor Count
Let's dive into a fun math problem involving factors of numbers, specifically the number 50. Filiz has observed that the number 50 has 'a' odd natural number factors and 'b' even natural number factors. Based on this information, we need to determine which of the given options is incorrect. This involves finding the factors of 50, categorizing them into odd and even factors, and then checking which of the provided equations or relations doesn't hold true.
Understanding the Factors of 50
First, let's find all the factors of 50. A factor of a number is any integer that divides the number evenly, leaving no remainder. To find the factors of 50, we can start by listing pairs of numbers that multiply to give 50.
- 1 x 50 = 50
- 2 x 25 = 50
- 5 x 10 = 50
So, the factors of 50 are: 1, 2, 5, 10, 25, and 50. Now, let's categorize these factors into odd and even numbers.
Odd Factors
Odd factors are those that are not divisible by 2. From the list above, the odd factors of 50 are 1, 5, and 25. Therefore, Filiz stated that a represents the number of odd factors, so a = 3.
Even Factors
Even factors are those that are divisible by 2. From the list of factors, the even factors of 50 are 2, 10, and 50. Filiz mentioned that b represents the number of even factors, so b = 3.
Verifying the Options
Now that we know a = 3 and b = 3, let's check each of the given options to see which one is incorrect.
A) a â‹… b = 9
Substituting the values, we have 3 â‹… 3 = 9. This statement is true.
B) a + b = 6
Substituting the values, we have 3 + 3 = 6. This statement is also true.
C) ab = 27
Substituting the values, we have 3<sup>3</sup> = 27 (since 3 x 3 x 3 = 27). This statement is true as well.
D) ba = 9
Substituting the values, we have 3<sup>3</sup> = 9. However, 3<sup>3</sup> = 3 x 3 x 3 = 27, not 9. Thus, this statement is false.
Therefore, the incorrect statement is option D: ba = 9.
Conclusion
In summary, we found that the number 50 has three odd factors (1, 5, 25) and three even factors (2, 10, 50). By substituting these values into the given options, we determined that the statement ba = 9 is false because 33 equals 27, not 9. Therefore, the correct answer is D.
Let's Explore Factors, Odd and Even!
Hey guys! Let's break down a cool math question together. Our friend Filiz is playing with numbers, and she's noticed something interesting about the number 50. She says that 50 has a certain number of odd factors (we'll call that a) and a certain number of even factors (we'll call that b). Our mission, should we choose to accept it, is to figure out which of the statements about a and b is a big, fat lie! So, buckle up, grab your thinking caps, and let's dive in!
What Are Factors, Anyway?
Before we start hunting for odd and even factors, let's make sure we're all on the same page about what a factor actually is. Simply put, a factor is a number that divides evenly into another number. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because each of those numbers divides into 12 without leaving a remainder. Got it? Great!
Cracking the Case of 50's Factors
Alright, let's get our hands dirty and find all the factors of 50. We can do this by systematically checking which numbers divide evenly into 50. Here's how it breaks down:
- 1 goes into 50 fifty times (1 x 50 = 50)
- 2 goes into 50 twenty-five times (2 x 25 = 50)
- 5 goes into 50 ten times (5 x 10 = 50)
And that's it! We've found all the factors of 50: 1, 2, 5, 10, 25, and 50. Now comes the fun part: sorting them into odd and even categories.
Oddballs and Even Stevens
So, what's the difference between an odd and an even number? Even numbers are divisible by 2, while odd numbers are not. Easy peasy! Let's apply this to our list of factors.
- Odd Factors of 50: 1, 5, and 25. These numbers can't be divided evenly by 2.
- Even Factors of 50: 2, 10, and 50. These numbers are all divisible by 2.
According to Filiz, a represents the number of odd factors, and b represents the number of even factors. So, a = 3 (because there are three odd factors) and b = 3 (because there are three even factors).
Time to Play Detective
Now that we know the values of a and b, we can put on our detective hats and see which of the given statements is false. Let's go through them one by one:
A) a â‹… b = 9
Is this true? Well, 3 â‹… 3 = 9. So, this statement is correct.
B) a + b = 6
Let's see... 3 + 3 = 6. Yep, this statement is also correct.
C) ab = 27
Okay, this one's a little trickier. *a*<sup>*b*</sup> means 3 raised to the power of 3, or 3 x 3 x 3, which equals 27. So, this statement is correct too!
D) ba = 9
Aha! Let's try this one. *b*<sup>*a*</sup> means 3 raised to the power of 3, which, as we just figured out, equals 27, *not* 9. **Busted!** This statement is the false one.
Case Closed!
So, there you have it, folks! By carefully examining the factors of 50 and using a little bit of detective work, we've determined that the incorrect statement is D) ba = 9. Great job, team! We've cracked the case of the odd and even factors.
Unraveling Factors: A Deep Dive into Number 50
In this mathematical exploration, our goal is to dissect the number 50, identifying its factors and categorizing them based on whether they are odd or even. The problem presents a scenario where Filiz has counted a odd and b even factors of 50, and we must determine which of the given statements regarding a and b is incorrect. This requires a systematic approach to finding all factors of 50, differentiating between odd and even numbers, and then verifying the accuracy of each provided option.
Step-by-Step Factorization of 50
The initial step in this problem is to identify all the factors of 50. Factors are integers that divide 50 without leaving any remainder. These factors can be found by determining pairs of numbers that, when multiplied together, yield 50. Let's list them out:
- 1 × 50 = 50
- 2 × 25 = 50
- 5 × 10 = 50
Therefore, the comprehensive list of factors for 50 includes 1, 2, 5, 10, 25, and 50. Next, we need to classify these factors into odd and even categories.
Differentiating Between Odd and Even Factors
Odd factors are defined as integers that are not divisible by 2, while even factors are divisible by 2. Applying this definition, we can categorize the factors of 50 as follows:
- Odd Factors: 1, 5, and 25
- Even Factors: 2, 10, and 50
From this classification, we can determine that a, representing the number of odd factors, is 3, and b, representing the number of even factors, is also 3.
Validation of the Given Options
With the values of a and b established, we proceed to evaluate each of the given options to identify the incorrect statement:
A) a â‹… b = 9
Substituting the values, we get 3 â‹… 3 = 9, which is a true statement.
B) a + b = 6
Substituting the values, we get 3 + 3 = 6, which is also a true statement.
C) ab = 27
Substituting the values, we get 3<sup>3</sup> = 27, as 3 raised to the power of 3 is indeed 27. Thus, this statement is true.
D) ba = 9
Substituting the values, we get 3<sup>3</sup> = 9, but this is incorrect. As we established earlier, 3<sup>3</sup> equals 27, not 9. Therefore, this is the statement we're looking for.
Conclusion: Identifying the Incorrect Statement
In conclusion, after carefully evaluating all factors of the number 50 and categorizing them as either odd or even, we determined that the incorrect statement is D) ba = 9. This is because the correct computation of 33 yields 27, contradicting the statement. Thus, the comprehensive analysis of factors and their properties leads us to the final answer.
