Question

How many factors does a product of three distinct primes have? What about a product of $4$ distinct primes?
[4 marks]
[SCERT, Kerala]
[Number of Factors]

Answer 1

Suppose a number is a product of three different prime numbers say $a,b$ and $c$

Since,$a,b$ and $c$ all are prime numbers, all of them have $2$ factors.
Multiplying each factor of $a$ with each factor of $b$ will give us $2\times 2$(i.e. $4$) factors of $a\times b.$
[1 mark]

Now, multiplying all the factors of $c$ with the factors of $a\times b$ will give us the factor of $a\times b\times c.$
That means, number of factors of $a\times b\times c=4\times 2=8.$
[1 mark]

Similarly, a number is product of four prime numbers say $a,b,c$ and $d.$

Since, $d$ is a prime number it will have $2$ factor.
And as shown above, $a\times b\times c$ have $8$ factors.
[1 mark]

Multiplying all the factors of $d$ with the factor of $a\times b\times c$ will give us the factor of $a\times b\times c\times d$
Therefore, number of factor of $a\times b\times c\times d=8\times 2=16$.
[1 mark]