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Divisibility

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CAT 2025 Lesson : Divisibility - Concepts & Cheatsheet

Note: The video for this module contains a summary of all the concepts covered in the Divisibility 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.

6. Cheatsheet

1) List of divisibility rules

Number	Divisibility Rule
$2$	Last digit divisible by $2$
$3$	Sum of digits divisible by $3$
$4$	Last two digits divisible by $4$
$5$	Last digit is $0$ or $5$
$6$	Divisible by $2$ and $3$
$7$	Difference between sum of alternate sets of digits is divisible by $7$
$8$	Last three digits divisible by $8$
$9$	Sum of digits divisible by $9$
$10$	Last digit is $0$
$11$	Difference between sum of alternate sets of digits is divisible by $11$
$12$	Divisible by $3$ and $4$
$13$	Difference between sum of alternate sets of digits is divisible by $13$
$14$	Divisible by $2$ and $7$
$15$	Divisible by $3$ and $5$
$16$	Last $4$ digits divisible by $16$
$18$	Divisible by $2$ and $9$
$20$	Divisible by $4$ and $5$
$24$	Divisible by $3$ and $8$
$25$	Last $2$ digits divisble by $25$
$32$	Last 5 digits divisible by $32$
$100$	Last $2$ digits are $0$
$125$	Last $3$ digits divisible by $125$

2) For all composite numbers,
Divisibility Rule: Check for divisibility by each of the prime factors raised to their respective powers.
Remainder Rule: Apply Chinese Remainder theorem.

3) Where

p

is a prime number, greatest power of p that divides n! is the sum of quotients when

n

is successively divided by

p

.

4) Where

a

b

and

c

are prime numbers and

x = a^{p}b^{q}c^{r}

, to find the greatest power of x that divides n!,
(a) find the largest powers for each of

a

b

and

c

that can divide

n!

;
(b) divide these respective powers by

p

q

and

r

and write down the quotients.
(c) The least quotient is the highest power of

x

that can perfectly divide

n!

.

5) To find the last

n

digits in the product of certain numbers, we can simply multiply the last

n

digits of each of these numbers.

6) Cyclicity of units digit is
(a) $\bm{1}$ for

0

1

5

6

(b) $\bm{2}$ for

4

9

2

3

7

8

7

) Last

2

digits of

x^{n}

when

$x$ is a	Last $2$ digits
number that ends in $5$	If power is even, then $25$ . If power is odd, then $25$ if tens digit of base is even and $75$ if tens digit of base is odd.
number that ends in $0$	Last $2$ digits are $00$
multiple of $2$ but not $4$	Last $2$ digits of $x^{40k + 1} = x + 50$
multiple of $4$	Last $2$ digits of $x^{40k + 1} = x$
number that ends in $1$	Last digit is $1$ , tens digit is $U(\text{tens digit of} x \times \text{units digit of} n)$
number that ends in $3$ , $7$ or $9$	Raise it by a power so that the number ends in $1$ . Then, apply the above rule for 'number that ends in $1$ '.

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