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14:40
Concepts & Cheatsheet
Vignesh Srinivasan14:40
Module 8 of 8
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
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 is a prime number, greatest power of p that divides n! is the sum of quotients when is successively divided by .
4) Where , and are prime numbers and , to find the greatest power of x that divides n!,
(a) find the largest powers for each of , and that can divide ;
(b) divide these respective powers by , and and write down the quotients.
(c) The least quotient is the highest power of that can perfectly divide .
5) To find the last digits in the product of certain numbers, we can simply multiply the last digits of each of these numbers.
6) Cyclicity of units digit is
(a) for , , ,
(b) for ,
(c) for , , ,
) Last digits of when
| Number | Divisibility Rule |
|---|---|
| Last digit divisible by | |
| Sum of digits divisible by | |
| Last two digits divisible by | |
| Last digit is or | |
| Divisible by and | |
| Difference between sum of alternate sets of digits is divisible by | |
| Last three digits divisible by | |
| Sum of digits divisible by | |
| Last digit is | |
| Difference between sum of alternate sets of digits is divisible by | |
| Divisible by and | |
| Difference between sum of alternate sets of digits is divisible by | |
| Divisible by and | |
| Divisible by and | |
| Last digits divisible by | |
| Divisible by and | |
| Divisible by and | |
| Divisible by and | |
| Last digits divisble by | |
| Last 5 digits divisible by | |
| Last digits are | |
| Last digits divisible by |
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 is a prime number, greatest power of p that divides n! is the sum of quotients when is successively divided by .
4) Where , and are prime numbers and , to find the greatest power of x that divides n!,
(a) find the largest powers for each of , and that can divide ;
(b) divide these respective powers by , and and write down the quotients.
(c) The least quotient is the highest power of that can perfectly divide .
5) To find the last digits in the product of certain numbers, we can simply multiply the last digits of each of these numbers.
6) Cyclicity of units digit is
(a) for , , ,
(b) for ,
(c) for , , ,
) Last digits of when
| is a | Last digits |
|---|---|
| number that ends in | If power is even, then . If power is odd, then if tens digit of base is even and if tens digit of base is odd. |
| number that ends in | Last digits are |
| multiple of but not | Last digits of |
| multiple of | Last digits of |
| number that ends in | Last digit is , tens digit is |
| number that ends in , or | Raise it by a power so that the number ends in . Then, apply the above rule for 'number that ends in '. |