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Number System
Basics and Test of Divisibility
Natural Numbers
• The numbers 1,2,3………… are called Natural numbers
• It is denoted by N.
Whole Numbers
• All natural numbers along with zero.
• 0,1,2,3………………..
• It is denoted by W.
Integers
• The natural numbers , negative of natural numbers and zero.
• …………………..,-2,-1,0,1,2………………..
• It is denoted by Z.
Rational Number
• Number of the form p/q where q != 0.
• P and q are integers.
Irrational Number
• A nonterminating and no recurring decimal.
• Eg. π (Pi) we cannot write simple fraction equal to Pi.
• 22/7 = 3.1428571428571... Is close but not accurate.
Real Numbers
• Collection of rational and irrational numbers.
Even Numbers
• Number divisible by 2
• 2,4,20,100 etc.
Odd numbers
• Numbers which are not divisible by 2
• 3,5,7,53 etc.
Prime Numbers
• Number other than 1 that is divisible by 1 and itself.
• 3,5,7,11,13,17 etc.
Co-Primes or Relatively Primes
• Two or more numbers are said to be co-primes, if they have only 1 as a
common factor.
• Example: (4,5), (2,3), (8,9) etc.
Composite Numbers
• Number other than 1 which is not prime are composite.
• 2,4,9,15 etc.
Test Of Divisibility
Divisibility by 2
• If the unit digit is any of 0,2,4,6,8.
• 20,66,58,1114 etc.
Divisibility by 3
• If sum of the digits is divisible by 3.
• Eg.123
• Here sum of digits=1+2+3=6 which is divisible by 3 hence 123 is divisible
by 3.
• Similarly, 567=5+6+7=18 which is divisible by 3 hence 567 is divisible by 3.
Divisibility by 4
• If the number formed by last two digit is divisible by 4.
• Eg. 1234548
• Here last two digit form 48 which is divisible by 4 Hence 1234548 is divisible
by 4.
Divisibility by 5
• If its unit digit is either 0 or 5.
• Eg. 10,102365065 etc.
Divisibility by 6
• If it is divisible by both 2 and 3.
• Eg. 12,12336 etc
Divisibility by 8
• If the number formed by the last three digits of given number is divisible by
8.
• Eg. 54672
• Here last three digits 672 is divisible by 8 hence 54672 is divisible by 8.
Divisibility by 9
• If sum of digit is divisible by 9.
• Eg.5967=
• Here sum of digits is 5+9+6+7=27 which is divisible by 9 hence 5967 is
divisible by 9
Divisibility by 10
• If its last digit is 0.
• Eg. 4656510,64656560 etc. are divisible by 10
Divisibility by 11
• If difference of sum of its digits at odd places and sum of its digits at even
places is either 0 or a number divisible by 11.
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Eg.29435417
Here sum of digits at odd places=2+4+5+1=12
Sum of digits at even places=9+3+4+7=23
Here diff is 23-12 =11 which is divisible by 11.
Hence 29435417 is divisible by 11.