Question

In how many ways $4900$ can be expressed as a product of $2$ positive integers?

Answer 1

Hint: In this question, we need to find the number of ways in which the number given to us can be represented as the product of two positive integers. So, we will first do the prime factorization of the number and then will determine the number of ways they can be moulded so that the number $4900$ can be expressed as a product of positive integers.

Complete step-by-step solution:
Here, we need to express $4900$ as a product of two positive integers.
A composite number is a number that has at least one divisor other than $1$ and the number itself. In other words, a composite number is formed by multiplying two smaller positive numbers. A prime number is a natural number greater than $1$ that is not a product of two smaller natural numbers.
We know that $4900$ is a composite number.
Prime factorization is a method of finding prime numbers which multiply to make the original number.
In prime factorization, we start dividing the number by the first prime number $2$ and continue to divide by $2$ until we get a decimal or remainder. Then divide by $3,5,7,....$etc. until we get the remainder $1$ with the factors as prime numbers. Then write the numbers as a product of prime numbers.
Thus, prime factorization of $4900$ is,
$4900 = 2 \times 2 \times 5 \times 5 \times 7 \times 7$
This can be also written in exponential form as $4900 = {2^2} \times {5^2} \times {7^2}$.
Now, we can notice that there are two $2's$, two $5's$ and two $7's$ in the prime factorization of the number $4900$. So, to represent the number $4900$ as a product of two positive integers, we have to select the prime factors from the prime factorization of the number.
So, we can select $0$, $1$, or $2$ $2's$ from the prime factorization of $4900$. So, there are three options regarding the power of $2$ in $4900$.
Similarly, we can select $0$, $1$, or $2$ $5's$ from the prime factorization of $4900$. So, there are three options regarding the power of $5$ in $4900$.
Also, there are three options regarding the power of $7$ in $4900$.
Hence, the number of factors of $4900$ is $3 \times 3 \times 3 = {3^3} = 27$.
Now, we have to represent the number $4900$ as the product of two positive integers. This can be done by forming appropriate pairs of factors of the number. Now, we can express a number as a product of two positive integers in $\dfrac{n}{2}$ ways if the number is not a perfect square where n is the number of factors of the number. If the number is a perfect square, then we can express the number as a product of two primes in $\dfrac{{n + 1}}{2}$ ways if the number is a perfect square.
We know that $4900$ is a perfect square and there are a total of $27$ factors of the number.
So, it can be expressed as a product of two positive integers in $\dfrac{{27 + 1}}{2} = 14$ ways.

Note: In this question it is important to note here that a composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than $1$ and itself. Prime factorization is also known as prime decomposition. Every other number can be broken down into prime number factors, but the prime numbers are the basic building blocks of all the numbers.