Question

The number of ways in which the number 7056 can be resolved as a product of 2 factor is
A.64
B.24
C.46
D.23

Answer 1

Hint: First we will find the prime factorization of given numbers in the form of \[{a^p}{b^q}{c^r}\], where \[a\], \[b\], \[c\] are prime numbers and the \[p\], \[q\], \[r\] are natural numbers as their respective powers and then use the formula of the number of factors which can be expressed as \[\left( {p + 1} \right)\left( {q + 1} \right)\left( {r + 1} \right)\]. Then we will use the number of ways to express the number as a product of two numbers is exactly half its number of factors by adding 1.

Complete step-by-step answer:
We are given that the number is 7056.
We will find the prime factorization of given numbers in the form of \[{a^p}{b^q}{c^r}\], where \[a\], \[b\], \[c\] are prime numbers and the \[p\], \[q\], \[r\] are natural numbers as their respective powers.
Rewriting the above number using the prime factorization, we get
\[ \Rightarrow 7 \times 7 \times 3 \times 3 \times 2 \times 2 \times 2 \times 2\]
We know that the prime factorization of given numbers in the form of \[{a^p}{b^q}{c^r}\], where \[a\], \[b\], \[c\] are prime numbers and the \[p\], \[q\], \[r\] are natural numbers as their respective powers.
Simplifying the above expression as powers, we get
\[ \Rightarrow {7^2} \times {3^2} \times {2^4}\]
We know that the number of factors which can be expressed as \[\left( {p + 1} \right)\left( {q + 1} \right)\left( {r + 1} \right)\].
Finding the value of e \[p\], \[q\], \[r\] in the above expression, we get
\[p = 2\]
\[q = 2\]
\[r = 2\]
Substituting these above values in the formula for number of factors, we get
\[
   \Rightarrow \left( {2 + 1} \right)\left( {2 + 1} \right)\left( {4 + 1} \right) \\
   \Rightarrow 3 \times 3 \times 5 \\
   \Rightarrow 45 \\
 \]
We know that the number of ways to express the number as a product of two numbers is exactly half its number of factors by adding 1.

Using this in the above value, we get
\[
   \Rightarrow \dfrac{{45 + 1}}{2} \\
   \Rightarrow \dfrac{{46}}{2} \\
   \Rightarrow 23 \\
 \]
Therefore, the number of ways of writing it as a product of 2 factors is 23.
Hence, option D is correct.

Note: In solving these types of questions, you should know that the prime factorization means finding the prime number, which gets multiplied together to form the original number. We need to follow the steps properly to avoid calculation mistakes. While solving these types of questions, some students take the prime factors of the given number as the divisors, which is wrong. We will find the remainders by taking the divisors separately.