Question

For some integer $$m$$, every even integer is of the form:
(a) $$m$$ (b) $$m+1$$ (c) $$2m$$ (d) $$2m+1$$

Answer 1

Hint:
Here, we need to check which of the given options represents every even integer where $$m$$ is some integer. We will check all the options for some odd and even value of the integer $$m$$. The option which represents every even integer for both the odd and even value of the integer, that value of $$m$$ is the correct option.

Complete step by step solution:
We will check each option for some odd or even value of the integer $$m$$ to find the correct option.
(a)
Suppose that $$m$$ is the integer 1.
The integer 1 is an odd integer.
Therefore, $$m$$ does not represent every even integer.
This is true for all values of $$m$$ that are odd integers.
Thus, we can conclude that if $$m$$ is any odd integer, then $$m$$ does not represent every even integer.
Therefore, option (a) is incorrect.
(b)
Suppose that $$m$$ is the integer 2.
The integer 2 is an even integer.
1 more than the integer 2 is $$2+1=3$$.
Therefore, if $$m=2$$, then $$m+1=3$$.
The number $$m+1$$ is an odd integer.
Thus, $$m+1$$ does not represent every even integer.
This is true for all values of $$m$$ that are even integers.
Thus, we can conclude that if $$m$$ is any even integer, then $$m+1$$ does not represent every even integer.
Therefore, option (b) is incorrect.
(c)
Suppose that $$m$$ is the integer 1.
The integer 1 is an odd integer.
Multiplying the integer 1 by 2, we get
$$1\times 2=2$$
Therefore, if $$m=1$$, then $$2m=2$$.
Here, the number $$2m$$ is an even integer.
Now, suppose that $$m$$ is the integer 2.
The integer 2 is an even integer.
Multiplying the integer 2 by 2, we get
$$2\times 2=2$$
Therefore, if $$m=2$$, then $$2m=2$$.
Here, the number $$2m$$ is an even integer.
We can observe that for any odd or even value of $$m$$, the value of $$2m$$ is an even integer.
Thus, we can conclude that if $$m$$ is any integer (even or odd), then $$2m$$ represents every even integer.
Therefore, option (c) is the correct option.
(d)
We have proved that if $$m$$ is any integer (even or odd), then $$2m$$ represents every even integer.
Therefore, since $$2m$$ is an even integer, then $$2m+1$$ represents every odd integer.

Thus, option (d) is incorrect.

Note:
We have proved that if $$m$$ is any integer (even or odd), then $$2m$$ represents every even integer. This is because when any integer is multiplied by 2, the resulting number has either 2, 4, 6, 8, or 0 as the digit in the unit’s place. Any number that has the digit 2, 4, 6, 8, or 0 in the unit’s place is divisible by 2, and every number divisible by 2 is an even number. Odd numbers are not divisible by 2 whereas even numbers are divisible by 2.