Question

Find the number of factors of \[1728\].
A) \[28\]
B) \[18\]
C) \[36\]
D) \[26\]

Answer 1

Hint: Here we can solve this question by using the formula of finding the total number of factors for any natural number by making it into the product of prime numbers by using the prime factorization method, which states that, If \[{\text{N}}\] is any natural number for which we need to find the number of factors then we will convert that number into the product of prime numbers as shown below:
\[{\text{N}} = {{\text{X}}^{\text{a}}} \times {{\text{Y}}^{\text{b}}} \times {{\text{Z}}^{\text{c}}}\] , where
\[{\text{X}}\], \[{\text{Y}}\] and \[{\text{Z}}\] are prime numbers and \[{\text{a}}\], \[{\text{b}}\] and \[{\text{c}}\] are their respective powers.
The formula for calculating the total number of factors for \[{\text{N = }}\]\[\left( {a + 1} \right)\left( {b + 1} \right)\left( {c + 1} \right)\]

Complete step-by-step solution:
Step 1: We need to find the number of factors of a number \[1728\], so by using the prime factorization method we will write all the prime factors of the number \[1728\] as shown below:
\[{\text{Prime factors of }}1728 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3\]
By writing the above factors into the form of powers, we get:
\[ \Rightarrow {\text{Prime factors of }}1728 = {2^6} \times {3^3}\]
Step 2: By using the formula of finding the factors of any natural number which is \[{\text{Number of factors}} = \left( {a + 1} \right)\left( {b + 1} \right)\left( {c + 1} \right)\] , we get:
\[ \Rightarrow {\text{Number of factors of }}1728 = \left( {6 + 1} \right) \times \left( {3 + 1} \right)\], where \[a = 6\] and \[b = 3\].
By solving inside the brackets of the above expression we get:
\[ \Rightarrow {\text{Number of factors of }}1728 = 7 \times 4\]
By doing the multiplication into the RHS side, we get:
\[ \Rightarrow {\text{Number of factors of }}1728 = 28\]

\[\therefore \] \[{\text{Number of factors of }}1728 = 28\]

Note: Students can also solve these questions by writing the factors of the given number and after that counting them.
\[{\text{Factors of }}1728 = 1\],
\[2\], \[3\], \[4\], \[6\], \[8\], \[9\], \[12\], \[16\], \[18\], \[24\], \[27\], \[32\], \[36\], \[48\], \[54\], \[64\], \[72\], \[96\], \[108\], \[144\], \[192\], \[216\], \[288\], \[432\], \[576\], \[864\], \[1728\].
So, the total number of factors \[1728\] is \[28\].