M is an odd integer. For each of the following numbers, check if the number is odd.
2m- 1
2m +1
m^2 - m
m^2 +m+1
Everyone will be odd but m^2 - m.
2 m is even ... multiplication by an even quantity
... so the 1st two numbers are odd
odd squared is odd
... odd minus (or plus) odd is even
... 3rd number is even
odd plus odd is even
... even plus odd is odd
... 4th number is odd
Oh, odd numbers, my specialty! Alright, let's check if these numbers are odd:
For 2m - 1: Well, if M is an odd integer, then 2m will always be an even number. And subtracting 1 from an even number always gives you an odd number. So yes, 2m - 1 is odd.
For 2m + 1: Again, since M is odd, 2m will be even. And when you add 1 to an even number, you get another odd number. So yes, 2m + 1 is odd too.
For m^2 - m: Well, we know that M is odd, so let's plug in an odd integer. Say M = 3. Then m^2 - m would be 3^2 - 3, which is 9 - 3, giving us 6. And 6 is even, not odd. So m^2 - m is not necessarily odd.
For m^2 + m + 1: Let's plug in another odd integer, say M = 5. Then m^2 + m + 1 would be 5^2 + 5 + 1, which is 25 + 5 + 1, giving us 31. And 31 is definitely an odd number. So, m^2 + m + 1 is odd.
To summarize:
2m - 1 is odd.
2m + 1 is odd.
m^2 - m is not necessarily odd.
m^2 + m + 1 is odd.
I hope that clears things up! Let me know if you need any more odd or funny answers!
To check if each of the given numbers is odd, we need to substitute the value of M into the expressions and determine if the resulting numbers are odd. Let's go step by step:
1. 2m - 1:
Substitute M as an odd integer: 2M - 1
Since 2 times an odd number is always even and subtracting 1 from an even number will always yield an odd number, we can conclude that 2M - 1 is an odd number.
2. 2m + 1:
Substitute M as an odd integer: 2M + 1
Again, since 2 times an odd number is always even and adding 1 to an even number will always yield an odd number, we can conclude that 2M + 1 is an odd number.
3. m^2 - m:
Substitute M as an odd integer: (M^2) - M
For any odd integer, squaring it will always result in an odd number. Subtracting an odd number from an odd number will always result in an even number. Therefore, (M^2) - M is an even number, not odd.
4. m^2 + m + 1:
Substitute M as an odd integer: (M^2) + M + 1
Similarly, squaring an odd integer will always result in an odd number. Adding an odd number to an odd number will always result in an even number. But then adding 1 to an even number will yield an odd number. Therefore, (M^2) + M + 1 is an odd number.
To summarize the results:
- 2M - 1 is odd.
- 2M + 1 is odd.
- (M^2) - M is even.
- (M^2) + M + 1 is odd.
To determine if a number is odd, we need to check if it is divisible by 2. If a number is not divisible by 2, then it is odd.
Let's go through each of the given expressions and see if they are odd:
1. 2m - 1:
To check if this number is odd, we need to substitute the value of m into the expression and then see if the result is divisible by 2. Since M is given as an odd integer, let's say M = 2k + 1, where k is an integer (according to the definition of odd numbers). Now, substitute this value into the expression:
2m - 1 = 2(2k + 1) - 1 = 4k + 2 - 1 = 4k + 1
Since 4k + 1 is not divisible by 2, the number 2m - 1 is odd.
2. 2m + 1:
Similarly, substitute M = 2k + 1 into the expression:
2m + 1 = 2(2k + 1) + 1 = 4k + 2 + 1 = 4k + 3
Since 4k + 3 is not divisible by 2, the number 2m + 1 is also odd.
3. m^2 - m:
Using the value of M = 2k + 1:
m^2 - m = (2k + 1)^2 - (2k + 1) = 4k^2 + 4k + 1 - 2k - 1 = 4k^2 + 2k
Since 4k^2 + 2k is divisible by 2, the number m^2 - m is not odd.
4. m^2 + m + 1:
Again, substitute M = 2k + 1 into the expression:
m^2 + m + 1 = (2k + 1)^2 + (2k + 1) + 1 = 4k^2 + 4k + 1 + 2k + 1 + 1 = 4k^2 + 6k + 3
Since 4k^2 + 6k + 3 is not divisible by 2, the number m^2 + m + 1 is odd.
To summarize:
- The number 2m - 1 is odd.
- The number 2m + 1 is odd.
- The number m^2 - m is not odd (even).
- The number m^2 + m + 1 is odd.