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Divisibility

Divisibility

MODULES

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Divisibility - 2, 5, 10
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Divisibility - 3, 9, 11
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7, 13 and Composite
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Divisibility of Factorial
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Last Digit
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Last 2 Digits
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Past Questions

CONCEPTS & CHEATSHEET

Concept Revision Video

SPEED CONCEPTS

Divisibility 1
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Divisibility 2
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PRACTICE

Divisibility : Level 1
Divisibility : Level 2
Divisibility : Level 3
ALL MODULES

CAT 2025 Lesson : Divisibility - Concepts & Cheatsheet

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Note: The video for this module contains a summary of all the concepts covered in this 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
222 Last digit divisible by 222
333 Sum of digits divisible by 333
444 Last two digits divisible by 444
555 Last digit is 000 or 555
666 Divisible by 222 and 333
777 Difference between sum of alternate sets of digits is divisible by 777
888 Last three digits divisible by 888
999 Sum of digits divisible by 999
101010 Last digit is 000
111111 Difference between sum of alternate sets of digits is divisible by 111111
121212 Divisible by 333 and 444
131313 Difference between sum of alternate sets of digits is divisible by 131313
141414 Divisible by 222 and 777
151515 Divisible by 333 and 555
161616 Last 444 digits divisible by 161616
181818 Divisible by 222 and 999
202020 Divisible by 444 and 555
242424 Divisible by 333 and 888
252525 Last 222 digits divisble by 252525
323232 Last 5 digits divisible by 323232
100100100 Last 222 digits are 000
125125125 Last 333 digits divisible by 125125125


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 ppp is a prime number, greatest power of p that divides n! is the sum of quotients when nnn is successively divided by ppp.

4) Where
aaa, bbb and ccc are prime numbers and x=apbqcrx = a^{p}b^{q}c^{r}x=apbqcr, to find the greatest power of x that divides n!,
(a) find the largest powers for each of
aaa, bbb and ccc that can divide n!n!n! ;
(b) divide these respective powers by
ppp, qqq and rrr and write down the quotients.
(c) The least quotient is the highest power of
xxx that can perfectly divide n!n!n!.

5) To find the last
nnn digits in the product of certain numbers, we can simply multiply the last nnn digits of each of these numbers.

6) Cyclicity of units digit is
(a)
1\bm{1}1 for 000, 111, 555, 666
(b)
2\bm{2}2 for 444, 999
(c)
4\bm{4}4 for 222, 333, 777, 888

777) Last 222 digits of xnx^{n}xn when

xxx is a Last 222 digits
number that ends in 555 If power is even, then 252525. If power is odd, then 252525 if tens digit of base is even and 757575 if tens digit of base is odd.
number that ends in 000 Last 222 digits are 000000
multiple of 222 but not 444 Last 222 digits of x40k+1=x+50x^{40k + 1} = x + 50x40k+1=x+50
multiple of 444 Last 222 digits of x40k+1=xx^{40k + 1} = xx40k+1=x
number that ends in 111 Last digit is 111, tens digit is U(tens digit ofx×units digit ofn)U(\text{tens digit of} x \times \text{units digit of} n)U(tens digit ofx×units digit ofn)
number that ends in 333, 777 or 999 Raise it by a power so that the number ends in 111. Then, apply the above rule for 'number that ends in 111'.
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