Question

The multiplicative identity for the integers is
A. 0.
B. -1.
C. 1.
D. None of these.

Answer 1

Hint: To solve this question, we have to know that the multiplicative identity is an identity which when used to multiply with a given element in a specified set such that the element remains unchanged, i.e. $a \times e = e \times a = a$ where e is the multiplicative identity and a is any element.

Complete step-by-step answer:
As we know that,
The multiplicative identity states that any number multiplied to its multiplicative identity gives the number itself.
We have to find the multiplicative identity for the integers.
So,
Let us say ‘e’ be the multiplicative identity of any integer ‘a’.
As according to the definition of multiplicative identity,
$a \times e = e \times a = a$ ………. (i)
Let us assume the value of a = 0, as 0 is an integer.
$0 \times e = e \times 0 = 0$
Here, we know that if we multiply any number with 0, we will get the answer 0.
So, we cannot find the multiplicative identity element with 0.
Let a = 1, putting in equation (i)
$ \Rightarrow 1 \times e = e \times 1 = 1$
This can be written as:
$ \Rightarrow 1 \times e = 1$,
Solving this, we will get
$ \Rightarrow e = 1$
Now, we will check this for negative integers.
Let a = -2, put in equation (i),
$ \Rightarrow - 2 \times e = e \times - 2 = - 2$
$ \Rightarrow - 2 \times e = - 2$
Solving this, we will get
$ \Rightarrow e = 1$
Therefore, we can say that the multiplicative identity for the integers is 1.
Hence, the correct answer is option (C).

Note:In order to solve this type of problems the key is to know the definition of multiplicative identity. It is also called the Identity property of multiplication, because the identity of the number remains the same. The identity element for multiplication operation will be always 1 and for addition operation, it will be always 0.
In the same way we solved above, even if we considered any algebraic term and applied the condition to it, we get the multiplicative identity to be 1 itself.