Numbers

Let's start with numbers.






How to check Divisibility by a number.

Divisibility check with 1:-

Just kidding, you know every number is divisible by 1.

Divisibility check with 2 :-

If the number ends with even digit, then it is divisible by 2.

Eg - 2,34,66 etc

Divisibility check with 3 :-

If sum of digits in a number is a multiple of 3,then the number is a multiple of 3.

Eg - 345 -> 3+4+5=12 , we know 12 is a multiple of 3, hence 345 is divisble by 3.

Divisibilty check with 4 :-

For a nummber to  be divisible by 4, then last 2 digits of the number must be divisible by 4.

Eg- 2584, Last 2 digits is 84 and it is divisble by 4, hence 2584 is a multiple of 4.

Divisibility check with 5 :-

If the number ends with 0 or 5 , then it is completely divisible by 5.

Eg - 8560,8565 etc are 5's multiple.

Divisibility check with 6 :-

For a number to be divisble by 6, then it must be divisible by both 2 and 3.

Eg - 48, divisible by 2 and 3.

Divisibilty by 8 :-

Last 3 digits must be divisible by 8.

Eg - 47744 , Take 744, Since 744 is divisible by 8 47744 is also divisible by 8.

Divisibility check with 9 :-

If the sum of digits of the number is divisible by 9 then, that number is a multiple of 9.

Eg - 153 , 1+5+3=9 ,so 153 is divisible by 9.

Divisibility check with 10 :-

If the number ends with 0, then it is a multiple of 10.

Eg - 20,30,630 etc

Divisibility check with 11 :-

Sum of digits of even positions and odd positions are substracted, the result must be 0 or multiple of 11.

Eg - 108416 ,
Sum of digits at even positions 1+8+1=10 ,
Sum of digits at odd positions 6+4+0=10,
And 10-10=0, hence 108416 is divisible by 11.

Divisibilty checck by 12 :-

The number should be divisible by both 3 and 4.(co-prime factors.)

Eg - 1152, is divisble by 3 and 4 ,hence divisble by 12.

Divisiblty check by 14,16,18, etc :-

Check with co-prime factors of the divisor.

Coming to some shortcut methods to check divisbilty by 7,13,17,19,29.

Remeber these numbers in order 7,13,29,19,17

Divisibility check with 7 :-

Take the digits till ten's place and with 5 multiplied with unit place digit. Check if it is divisibe by 7.

Eg :- Consider 455, 45+(5*5)=70 which is divisible by 7

Divisibilty check with 13 :-

Multiply unitplace digit with 4 and add ten's digits to it.If the result is a multiple of 13,then the number is also divisible by 13.
 Eg :- Consider 1248 - 124+(8*4)=156 is 13's multiple,hence 1248 is divisible  by 13.

Divisibilty check with 29 :-

Multiply unitplace digit with 3 and add ten's digits to it.If the result is a multiple of 29,then the number is also divisible by 29.
 Eg :- Consider 1624 - 162+(4*3)=174 is 29's multiple,hence 1624 is divisible  by 29.

Divisibilty check with 19 :-

Multiply unitplace digit with 2 and add ten's digits to it.If the result is a multiple of 19,then the number is also divisible by 19.
 Eg :- Consider  - 1235 - 123+(5*2)=133 is 19's multiple,hence 1235 is divisible  by 19.

Divisibilty check with 17 :-

Multiply unitplace digit with -5 and add ten's digits to it.If the result is a multiple of 17,then the number is also divisible by 17.
 Eg :- Consider  - 1105 - 110+(5*-5)=85 is 17's multiple,hence 1105 is divisible  by 17.








So let's talk about HCF and LCM
HCF - Highest Common Factor
LCM - Least COmmon Multiple
First we will see how to find LCM and HCF,then we will go to some tips and tricks to questions from these kinds.
How to find HCF of two numbers:-

Lets take two numbers,say 36 and 81

Step 1 - Split to it's factors =>
36=2*2*3*3
81=3*3*3*3

Step 2 - Take common factors =>
Here it is 3*3 = 9

Hence HCF of 36 and 81 is 9.

How to find LCM of two numbers :-

Let's take two numbers,say 36 and 81

Step 1 - Do factorization =>
36=2*2*3*3
81=3*3*3*3

Step 2 -Take every factor in it's largest power. =>
2*2 from 36
3*3*3*3 from 81

Multiply selected factors => 2*2*3*3*3*3=324

Hence 324 is the LCM of 36 and 81.

Let's take another example;

Take 18 and 30
18=2*3*3
30=2*3*5
Take factors in largest form,
LCM=2*3*3*5=90

Now come to important note in this topic;

HCF*LCM=Product of two choosen numbers.

Consider the above example and verify.
HCF of 36,81 = 9
LCM of 36,81 = 324

=>9*324=36*81