CAT 2025 Lesson : Factors & Remainders - Number of Factors

2. Number of Factors

Any natural number represented in the form $a^p \times b^q \times c^r \times ...$, where $a$, $b$, $c$, ... are prime numbers has ($p + 1$) ($q + 1$) ($r + 1$) ... factors that are natural numbers..

Note: For every positive factor of a number, there exists a negative factor. For instance, the positive factors perfectly divisible by $6$ are $1, 2, 3$ and $6$ and the negative factors that divide $6$ are $-1, -2, -3$ and $-6$.

The formulae provided in this section are for positive integers (natural numbers) only. To find the number of integral factors (i.e. positive and negative), you can multiply the number of positive factors by $2$.

2.1 Odd and Even factors

Product of odd numbers is always odd. Therefore, to calculate the number of odd factors that are positive, we ignore $2^n$ and apply the formula given for number of factors.

Even Factors $=$ Total Factors $-$ Odd Factors

Example 9

How many odd and even positive integral factors exist for $60$?

Solution

$60 = 2^2 \times 3^1 \times 5^1$
To find out the positive odd factors, we ignore $2^2$.
Number of factors of $3^1 \times 5^1 = (1 + 1) \times (1 + 1) = 4$

∴ Odd factors of $60$ $= \bm{4}$

Even factors of $60 =$ Total Factors $-$ Odd Factors $= 12 - 4 = \bm{8}$

Answer: $4$ and $8$

2.2 Expressing as product of two numbers

If a natural number $\bm{x}$ has $\bm{n}$ factors, then $x$ can be expressed as a product of two natural numbers in

1) $\dfrac{n}{2}$ ways if $n$ is even; and

2) $\dfrac{(n + 1)}{2}$ ways if $n$ is odd.

Number	Factors	Product of Numbers
$4$	$1, 2, 4$	$(1 \times 4), (2 \times 2)$
$6$	$1, 2, 3, 6$	$(1 \times 6), (2 \times 3)$
$8$	$1, 2, 4, 8$	$(1 \times 8), (2 \times 4)$
$9$	$1, 3, 9$	$(1 \times 9), (3 \times 3)$

When $\bm{n}$ is even, there are $\dfrac{n}{2}$ ways of expressing $x$ as a product of $2$ natural numbers.

Where there are an odd number of factors, the middle term multiplied with itself is to be counted (like in the case of numbers $4$ and $9$ in the table above). However, the middle term is written only once.

∴ When $\bm{n}$ is odd, there are $\dfrac{(n + 1)}{2}$ ways of expressing $x$ as a product of $2$ natural numbers.

2.3 Product of Factors

If a natural number $x$ has $n$ positive factors, then the product of its positive factors is $x^{\left (\frac{n}{2}\right)}$.

This is an extension of the concept explained in 2.2 above.

Note that in case of odd factors, the middle term is being considered only once. Therefore, the formula $x^{\left (\frac{n}{2}\right)}$ applies where $n$ is odd as well.

Example 10

In how many different ways can $36$ be expressed as the product of two natural numbers and what is the product of all the positive factors of $36$?

Solution

$36 = 2^2 \times 3^2$

Number of factors $= (2 + 1)\times (2 + 1) = 9$

Number of ways to express as product of $2$ numbers $= \dfrac{(9 + 1)}{2} = 5$

Product of factors = $36^{\left (\frac{9}{2}\right) }= 6^9$

Answer: $5, 6^9$