Question

Write the cardinal number of the set of all integers x which \[\dfrac{{60}}{x}\] is a natural number.

Answer 1

Hint:- We had to find the factors of 60 because the result of any fractional number is the natural number only if the denominator is the factor of the numerator.

Complete step-by-step answer:
As we know that x will be a factor of any number y only if their y is perfectly divisible by x. i.e. in other words we can say that \[\dfrac{y}{x}\] is a natural number.
So, for \[\dfrac{{60}}{x}\] to be a natural number, x must be a factor of 60.
Now the factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.
So, there are a total 12 factors of 60 written above.
Hence, x can have 12 different values.
As we know that the cardinal number for a set is the total number of possible elements of x.
Here set containing values of x will have 12 elements.
Set = {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60}
Hence, the type of problem cardinal number of sets of all integers x will be 12.

Note:- Whenever we come up with this type of problem then first we have to find all possible factors of the numerator (here 60) and after that x will be equal to the number of factors because when we divide any number by its factor then the resultant is a natural number, and remember that factors do not mean prime factors because prime factors are those factors which are also a prime number (i.e. only divisible by 1 and itself).