Question

The number of positive divisors of 252 is _______.
A.9
B.5
C.18
D.10

Answer 1

Hint: The aim is to find the positive divisors of 252 so, as we know that if \[a = {p_1}^{{\alpha _1}} \cdot {p_2}^{{\alpha _2}} \cdot .....\] then the total number of positive divisors of \[a\] are obtained by \[T\left( a \right) = \left( {{\alpha _1} + 1} \right)\left( {{\alpha _2} + 1} \right).....\] where \[{p_1},{p_2},...\] represents the prime numbers. Now, we will consider the given number 252 for which we need to find the prime divisors, so we will find the factors in prime numbers form and the prime numbers have multiplicity. Thus, we will substitute the values in powers of prime numbers in the general formula and obtain the result.

Complete step by step answer:
We will first consider the given number that is 252.
We need to find the positive divisors of the given number.
So, as we know that if \[a = {p_1}^{{\alpha _1}} \cdot {p_2}^{{\alpha _2}} \cdot .....\] then the total number of positive divisors of \[a\] are obtained by \[T\left( a \right) = \left( {{\alpha _1} + 1} \right)\left( {{\alpha _2} + 1} \right).....\].
Thus, we will find the factors in prime numbers form,
We get,
\[ \Rightarrow 252 = {2^2} \times {3^2} \times {7^1}\] where 2, 3 and 7 are prime numbers.
Form the above expression, we have \[{\alpha _1} = 2,{\alpha _2} = 2,{\alpha _3} = 1\]
Now, we will substitute it in the expression, \[T\left( a \right) = \left( {{\alpha _1} + 1} \right)\left( {{\alpha _2} + 1} \right).....\].
Thus, we get,
\[
   \Rightarrow T\left( {252} \right) = \left( {2 + 1} \right)\left( {2 + 1} \right)\left( {1 + 1} \right) \\
   \Rightarrow T\left( {252} \right) = 3 \cdot 3 \cdot 2 \\
   \Rightarrow T\left( {252} \right) = 18 \\
 \]
Hence, we can conclude that there total 18 positive divisors of 252.
Thus, option C is correct.

Note: We can also find the divisors by finding the numbers which are divisible by 252 and hence list them out. They are 1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252.
In the above method done in the solution part, the factors should be the prime numbers. The divisors of positive integers are the integer that evenly divides it. We just have to consider the positive divisors and not the negative ones.