Divisibility
Number & their classification:-
Natural Number(N):- {1, 2, 3, 4, 5, …………….}
Whole Number(W):- {0, 1, 2, 3, 4, 5, ……………. }
Integer(Z):- {……. -4, -3, -2, -1, 0, 1, 2, 3, ……….}
Rational Number(Q):- \(\frac{1}{2}\), \(\frac{3}{7}\), \(\frac{1}{2}\), \(\frac{-6}{5}\), 0, +5, -130, ………..
Irrational Number:- π, \(\sqrt 2 \), \(\sqrt 5 \), \(\sqrt 7 \), …………
Complex Number:-\(\sqrt { – 8} \), \(\sqrt { – 7} \), \(\sqrt { – 5} \), \(\sqrt { – 1} \), \(2 + \sqrt { – 5} \), \(3 + \sqrt { – 7} \), …………
Test for divisibility of numbers
Divisibility by 2:- If the last digit is an even number or it has 0 at the end then the number is divisible by 2.
Ex: 20, 50, 512, 678, 980324, 278
Divisibility by 3:- If the sum of the digits of the given number is divisible by 3 then the number is divisible by 3
Divisibility by 4:- If the number formed by last two digits of a number is divisible by 4 then the number is divisible by 4.
Divisibility by 5:- If the last digit of the given number is 0 or 5 then the number is divisible by 5.
Divisibility by 6:- If the given number is divisible by both 2 and 3 then the number is divisible by 6.
Divisibility by 7:-
Method (1):
Take the last digit of the number an multiply it by 2.
Subtract the result from the rest of the number
Repeat the process if necessary until you get a small number.
If the result is 0 or multiple of 7, the original number is divisible by 7 otherwise, it is not.
Method(2): Adding 5 times the last digit to the rest gives a multiple of 7.
Method(2): Adding 5 times the last digit to the rest gives a multiple of 7.
Method(3):
Using three-digit pairs:-
⟶ Split the number into groups of three digits from right to left.
⟶ calculate the alternating sum of these groups
⟶ If the result is 0 or divisible by 7, the original number is divisible by 7.
Divisibility by 8:- If the number formed by the last three digits of the given number is divisible by 8 then the given number is divisible by 8.
Divisibility by 9: If the sum of digits of the given number is divisible by 9.
Divisibility by 10:- If the last digit of the number is 0.
Ex: 680 57320 4100850
Divisibility by 11:- If the difference of the sum of its digits in odd places and the sum of its digits in even places is either 0 or multiplee by 11.
Divisibility by 13:-
Method (1): Add 4 times the last digit to the rest. The result must be 0 or divisible by 13.
Method(2): Subtract 9 times the last ddigit from the rest. The result must be 0 or divisible by 13.
Method(3):
⟶ Split the number into groups of three from right to left
⟶ calculate the alternating sum of these groups
⟶ If the result is 0 or divisible by 13, the original number is divisible by 13
Divisibility by 17:-
Method(1): Subtract 5 times the last digit from the rest. The result must be 0 or divisible by 17.
Method(2): Add 12 times the last digit to the rest. The result must be 0 or divisible by 17.
Some General Property of Divisibility:-
● If a number x is divisible by y , then any number divisible by x, will also be divisible by y and all the factors of y.
Ex: The number 72 is divisible by 6. Thus any number that is divisible by 72, will also be divisible by 6 an also factors of 6 i.e. by 2 and 3.
● If a number x is divisible by two or more than two co-prime numbers then x is also divisible by the product of those numbers.
Ex:- The number 2040 is divisible by 3, 8, 17 that are co-prime to each other so 2040 will also be divisible by 3 × 8 = 24, 3 × 17 = 51, 8 × 17 = 136
● If two numbers x and y are divisible by ‘p’, then their sum x + y is also divisible by p.
Ex:- The number 138 and 312 are both divisible by 6. Thus their sum = 138 + 312 = 450 will also be divisible by 6.
● If any number is written 6 times like 111111, 222222, 777777 etc, it will be exactly divisible by 7, 11, 13, 37 & 1001
● Any number written like 2727, 3535, 2929 will be divisible by 101.
● (aⁿ + bⁿ) n is odd then this number is divisible by (a + b)
● (aⁿ + bⁿ) n is even then it can not be determined.
● A six digit number is formed by repeating a three digit number; for example 257257, 689689, 649649 then this form of number is exactly divisible by 7, 11, 13, 1001.
● If a is divisible by b then ac is also divisible by b.
● If a is divisible by b and b is divisible by c then a is divisible by c.
● If two numbers a & b are divisible by a third number c then (a – b) is also divisible by c.
Successive Division:- successive division, also known as repeated division, involves dividing a number by a sequence of divisors, where quotient from one divisor becomes the dividend for the next. This process continues until a quotient of zero is obtained.
Ex:- Successive division of a number by 2 3, 5 and 7 gives remainder 1, 4, 0 and 5 respectively. What will be the sum of remainders if same number is divided by 7, 5, 3 and 2 successively ?
Sol:
(Q). A certain number when successively divided by 4, 5 and 7 leaves remainder 2, 3, 4 respectively. Find such least number
(i) 374 (ii) 514 (iii) 234 (iv) 92
Sol:
● Product of n consecutive number is always divisible by n!
Ex: n(n + 1)(n + 2)(n + 3)(n + 4) ⟶ divisible by 5!
(n – 3)(n – 2)(n – 1)(n)(n + 1)(n + 2) ⟶ divisible by 6!
(Q). n² – n is always divisible by which number.
Sol:
(n – 1)(n)(n + 1)
⟶ divisible by 3! = 6 Answer