The first statement is false. The second statement is true. Explain why, using words, pictures, or numbers.
If a number has 2 and 6 as factors, then it has 12 as a factor.
If a number has 12 as a factor, then both 2 and 6 are factors.
false. consider 18
true. If N = 12m then N = 2*6*m
so both 2 and 6 are factors
rebqtj34
Let me explain this with a little humor! Imagine the number as a person, and the factors as their friends.
So, in the first statement, if the number has 2 and 6 as factors, then let's picture it as a person (number) surrounded by their friends (factors). Here's what it looks like:
2 6
| |
Friend 1 - | Number |
Friend 2 - | |
| |
Isn't it cute? This group of friends represents the factors of the number.
In the second statement, it says that if the number has 12 as a factor, then both 2 and 6 must also be factors. Let's see how it looks now:
2 6
| |
Friend 1 - | Number |
Friend 2 - | |
Friend 3 - | 12 |
| |
Oh look, our number made a new friend named 12! And as per the statement, 12 is a factor of the number. But guess what? Our existing friends 2 and 6 are still there! So, the second statement is true.
Now, why is the first statement false? Let me show you:
2 6
| |
Friend 1 - | Number |
Friend 2 - | |
| |
Wait a minute! In the first statement, it claimed that if the number has 2 and 6 as factors, then it must have 12 as a factor. But in our picture, there's no friend named 12 around our number! So, the first statement is false.
In short, the second statement is true because if a number has 12 as a factor, then both 2 and 6 must also be factors. But the first statement is false because even if a number has 2 and 6 as factors, it does not necessarily mean that 12 is also a factor.
To explain the truth of the second statement, "If a number has 12 as a factor, then both 2 and 6 are factors," we can use logical reasoning and a simple example.
Let's assume we have a number, N, that has 12 as a factor. This means that N can be divided evenly by 12. In mathematical terms, we can represent this as: N = 12 * k, where k is any whole number.
Now, let's try to express N in terms of 2 and 6. We know that 12 can be expressed as the product of 2 and 6: 12 = 2 * 6.
By substituting this into our initial equation, we have: N = (2 * 6) * k.
Simplifying further, N = 2 * (6 * k) = 2 * m, where m is the product of 6 and k.
This shows that if a number has 12 as a factor, then it can be expressed as a multiple of 2. Therefore, both 2 and 6 are factors of the number.
Thus, the second statement, "If a number has 12 as a factor, then both 2 and 6 are factors," is true.
To explain why the first statement is false and the second statement is true, let's look at the concept of factors.
Factors are numbers that can be multiplied together to get a specific number. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
Now, let's examine the first statement: "If a number has 2 and 6 as factors, then it has 12 as a factor." This statement is false.
To verify this, let's try to find a number that has 2 and 6 as factors but does not have 12 as a factor. The number 6 itself is a good example since it has both 2 and 6 as factors, but 12 is not a factor of 6. This disproves the first statement.
Moving on to the second statement: "If a number has 12 as a factor, then both 2 and 6 are factors." This statement is true.
To confirm this, we need to find a number that has 12 as a factor and verify if both 2 and 6 are factors of that number. Let's consider the number 24.
We can see that 12 is a factor of 24 since 12 can be multiplied by 2 to get 24. Additionally, both 2 and 6 are factors of 24. This validates the second statement.
In summary, the first statement is false because there are numbers that have 2 and 6 as factors, but not 12. On the other hand, the second statement is true; if a number has 12 as a factor, then both 2 and 6 will also be factors of that number.