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Must Do: Divisiblity Concepts with Example Questions

Concept – Divisibility of a number


Divisibility by 2 — A number is divisible by 2 if the unit digit is zero or divisible by 2.
Ex. – 2, 4, 8, 16, 18 etc. are all divisible by 2.
Divisibility by 3 — A number is divisible by 3 if the sum of digits of the number is divisible by 3. 
Ex. – if 43n25 is divisible by 3 then find the minimum value n
Sol. 4+3+n+2+5 is divisible by 3 
So n=1,4,7
Minimum value of n=1 

Divisibility by 4 — A number is divisible by 4 if the number formed by the last two digits is divisible by 4 .
Ex. – 56248 is divisible by 4 since 48 is divisible by 4. 

Divisibility by 5 — A number is divisible by 5 if unit’s digit is 0 or 5. 
Ex. – 1225, 13750, 245 etc. are all divisible by 5. 

Divisibility by 7 — The unit digit of given number is doubled and then it is subtracted from the number obtained after omitting the unit digit. 
If the remainder is divisible by 7 then the given number is also divisible by 7.
 Ex. -is  343 divisible by 7 or not
Sol. Double unit digit 3 of 343 is 6 
Then, 34 – 6 = 28 which is divisible by 7.

Divisibility by 8 — A number is divisible by 8, if the number formed by the last 3 digits is divisible by 8.
Ex. 21056 is divisible by 8, as the number formed by last three digit i.e. 056 is divisible by 8. 

Divisibility by 9 — A number is divisible by 9 if the sum of its digits is divisible by 9.
 Ex. If 256x5 is divisible by 9, find the maximum  value of x 
Sol.   value of x divisible by 9 ⇒(18+x)/9
x=0,9       maximum value of x =9

Divisibility by 10 — An number is divisible by 10 if its last digit is zero 

 Divisibility by 11 — An number is divisible by 11 if the differences of the sum of the digit at odd places and sum of digit at even places is either zero or divisible by 11.
Ex. If 3x4567 is divisible by 11 then what is the value of x
Sol. 3+4+6-x+5+7=0 or divisible by 11
⇒1-x=0  
x=1  

Divisibility by 25 —  A number is divisible by 25 if the number formed by the last two digits is divisible by 25 or the last two digit are zero. 

Divisibility by 125 — A number is divisible by 125 if the number formed by last three digit is divisible by 125 or the last three digit are zero 

Divisibility by Composite number 
Divisibility of Composite number is the divisibility of its Coprime factor 
For Ex. – 
1. divisibility of 33 is divisibility of 3 and divisibility of 11 
2. divisibility of 165 is the divisibility of 3, 5 and 11. 

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