If n is an integer, which of the following must be odd?
a. 3n-5
b. 3n+4
c. 4n+10
d. 4n-5
e. 5n+7
please answer and explain
d, right
even+odd = odd
4n is even
so, what do you think?
correct
Well, let's put on our math clown nose and tackle this question!
To figure out which expressions must be odd, we need to remember that odd numbers are not divisible by 2. In other words, if we divide an odd number by 2, we won't get a whole number.
Let's go through each expression and see if they must be odd:
a. 3n-5: This expression will be odd if the value of 3n is even, since subtracting 5 won't change the oddness. For example, if n = 2, we have 3(2) - 5 = 1, which is odd. However, if n = 1, we have 3(1) - 5 = -2, which is even. So, this expression is not guaranteed to be odd.
b. 3n+4: This expression will be odd if the value of 3n is odd, since adding 4 to an odd number will always give you an even number. For example, if n = 2, we have 3(2) + 4 = 10, which is even. If n = 1, we have 3(1) + 4 = 7, which is odd. So, this expression is not guaranteed to be odd.
c. 4n+10: This expression will be odd if the value of 4n is even, since adding 10 won't change its evenness. For example, if n = 2, we have 4(2) + 10 = 18, which is even. If n = 1, we have 4(1) + 10 = 14, which is even. So, this expression is not guaranteed to be odd.
d. 4n-5: This expression will be odd if the value of 4n is even, since subtracting 5 won't change its evenness. For example, if n = 2, we have 4(2) - 5 = 3, which is odd. If n = 1, we have 4(1) - 5 = -1, which is odd. So, this expression is guaranteed to be odd.
e. 5n+7: This expression will be odd if the value of 5n is even, since adding 7 won't change its evenness. For example, if n = 2, we have 5(2) + 7 = 17, which is odd. If n = 1, we have 5(1) + 7 = 12, which is even. So, this expression is not guaranteed to be odd.
Therefore, the only expression that must be odd is d. 4n-5.
Hope that helps, and remember, humor is the prime factor of math!
To determine which of the given expressions must be odd for any integer value of n, we can use the property that odd numbers are not divisible by 2.
Let's examine each expression individually:
a) 3n - 5:
To check if this expression is always odd, we need to consider when it is divisible by 2. Assuming 3n - 5 is divisible by 2, we can set up the equation 3n - 5 = 2k, where k is an integer. By rearranging the equation, we have 3n = 2k + 5. Here, the right side is an odd number (2k is always even and adding 5 makes it odd), which means the left side (3n) must also be an odd number. Therefore, 3n - 5 must be odd.
b) 3n + 4:
Similarly, we can set up the equation 3n + 4 = 2k and rearrange it as 3n = 2k - 4. Here, the right side is even (2k is always even and subtracting 4 maintains its evenness), which means the left side (3n) must also be even. Therefore, 3n + 4 cannot be odd.
c) 4n + 10:
Setting up the equation 4n + 10 = 2k, we can rearrange it as 4n = 2k - 10. The right side is even since 2k is always even and subtracting 10 does not change its evenness. Therefore, the left side (4n) must also be even. Thus, 4n + 10 cannot be odd.
d) 4n - 5:
Setting up the equation 4n - 5 = 2k, we can rearrange it as 4n = 2k + 5. The right side is an odd number since 2k is always even and adding 5 makes it odd. Therefore, the left side (4n) must also be odd. Thus, 4n - 5 must be odd.
e) 5n + 7:
Setting up the equation 5n + 7 = 2k, we can rearrange it as 5n = 2k - 7. The right side is an odd number since 2k is always even and subtracting 7 makes it odd. Therefore, the left side (5n) must also be odd. Thus, 5n + 7 must be odd.
From the above analysis, the expressions that must be odd are:
a. 3n - 5
and
e. 5n + 7