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Factors & Remainders

ALL MODULES

CAT 2025 Lesson : Factors & Remainders - Concepts & Cheatsheet

Note: The video for this module contains a summary of all the concepts covered in the Factors & Remainders lesson. The video would serve as a good revision. Please watch this video in intervals of a few weeks so that you do not forget the concepts. Below is a cheatsheet that includes all the formulae but not necessarily the concepts covered in the video.

8. Cheatsheet

1

) When two or more numbers are prime factorised, HCF is the product of primes with the least powers across the numbers and LCM is the product of primes with the highest powers across the numbers. LCM

\geq

HCF

2

) For

2

numbers

a

and

b

a \times b

= HCF(

a

b

)

\times

LCM (

a

b

)

3

) HCF of fractions =

\dfrac{\mathrm{HCF \ of \ numerators}}{\mathrm{LCM \ of \ denominators}}

and LCM of fractions =

\dfrac{\mathrm{LCM \ of \ numerators}}{\mathrm{HCF \ of \ denominators}}

4

) Any 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.

4.1

) For

2^p \times b^q \times c^r \times .....

, where

b

c

, ... are odd prime numbers, odd number of factors = (

q

1

)(

r

1

) ... .

4.2

) Total Factors

-

Odd Factors = Even Factors

5

) If a number

\bm{x}

has

\bm{n}

factors, then x can be expressed as a product of two natural numbers in

\dfrac{n}{2}

ways if

n

is even and

\dfrac{(n + 1)}{2}

ways if

n

is odd.

6

) If a number

\bm{x}

has has

\bm{n}

factors, then the product of its factors is

x^{\left(\frac{n}{2}\right)}

7

) HCF Concepts
(a) The largest number that divides

x

y

and

z

to leave remainders

a

b

and

c

= HCF[(

x

a

), (

y

b

), (

z

c

)]

(b) The largest number that divides

x

y

and

z

to leave the same remainder = HCF[(

x

y

), (

y

z

)]

8

) LCM Concepts
(a) Numbers leaving the same remainder (

r

) for a given set of divisors =

\bm{k}

\times

LCM of divisors +

\bm{r}

(b) Numbers leaving the same difference (

d

) in remainders and divisors =

\bm{k}

\times

LCM of divisors -

\bm{d}

dq

r

format and find the smallest possible solution, say

x

. These numbers are of the form

\bm{k}

\times

LCM of divisors +

\bm{x}

9

) Successive Division: Once we find the smallest number (say

x

) that satisfies, other numbers are of the form

\bm{k}

\times

Product of divisors +

\bm{x}

10

) Remainder rules

\mathrm{Rem} \left(\dfrac{n}{d}\right) = \mathrm{Rem} \left(\dfrac{a_1 + a_2 + a_3 +......}{d}\right)

\mathrm{Rem} \left(\dfrac{a_1}{d}\right) + \mathrm{Rem} \left(\dfrac{a_2}{d}\right) + \mathrm{Rem} \left(\dfrac{a_3}{d}\right) + .....

\mathrm{Rem} \left(\dfrac{n}{d}\right) = \mathrm{Rem} \left(\dfrac{a_1 \times a_2 \times a_3 \times......}{d}\right)

\mathrm{Rem} \left(\dfrac{a_1}{d}\right) \times \mathrm{Rem} \left(\dfrac{a_2}{d}\right) \times \mathrm{Rem} \left(\dfrac{a_3}{d}\right) \times .....

\mathrm{Rem} \left(\dfrac{(dq + r)^n}{d}\right) = \mathrm{Rem} \left(\dfrac{r^n}{d}\right)

11

) Euler's Totient of

n [\phi(n)]

is the number of co-prime positive integers less than

n

.
If the prime factors of

n

are

x_1, x_2, x_3, ....

\bm{ \phi (n)}

n \times \left(1 - \dfrac{1}{x_1}\right) \times \left(1 - \dfrac{1}{x_2}\right) \times \left(1 - \dfrac{1}{x_3}\right) \times .....

12

) Fermat's Theorem: If

n

and

d

are co-prime integers, then

\mathrm{Rem} \left(\dfrac{n^{\phi(d)}}{d}\right) = 1

13

) Chinese Remainder: Where n and d are not co-prime, the remainders for each of the prime factors (raised to their respective powers) is found. These are then used to find the common remainder.

14

) Number of ways in which a number

n

can be expressed as the difference of squares of positive integers is

\dfrac{x}{2}

x

is even and

\dfrac{(x - 1)}{2}

x

is odd, where

(a)

x

is the number of factors of

n

\bm{n}

is odd; and
(b)

x

is the number of factors of

\dfrac{n}{4}

\bm{n}

is divisible by 4.

15

) Number of ways in which

x

can be expressed as the sum of consecutive natural numbers is the number of odd factors of

x

other than

1

16

) Where

k

is a constant, the remainder when a polynomial $f(x)$ is divided by (

x

k

), is

f(k)

17

) Wilson's Theorem: Where

n

is a prime number,

\mathrm{Rem} \left(\dfrac{(n - 2)!}{n}\right) = 1

and

\mathrm{Rem} \left(\dfrac{(n - 1)!}{n}\right) = n - 1

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