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Numbers and divisibility rule
H.C.F and L.C.M
Factors
Multiples
Remainders
Cyclicity
Factorials
Types of numbers
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Real numbers :
These numbers can be represented on
the number line. For ex: any no ranges from -∞ to +∞
Imaginary numbers:
These numbers can not be
represented on the number line. For ex: i = √-1
Complex numbers: these numbers include both real
and imaginary numbers.
Real numbers
Rational
numbers. E.g
1, 2, 2/3, 3/4
Integers. E.g ∞, -4, 0 10,
+∞
Whole
numbers. E.g
0, 1, 2, … +∞
Natural
numbers. E.g
1, 2,….+∞
Irrational
numbers. E.g
√2/3, √2, √7
Fractions. E.g
½, 2/3,
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Even numbers: these numbers are divisible by
2. Example: 2,4 ,6,…
Odd numbers: these numbers are not
divisible by 2. Example: 1,3,5,…
Prime numbers: these numbers does not have
a divisor apart from 1 and itself. Example:
2,3,5,11,29 etc.
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Composite numbers: the numbers except 1
which are not prime are called composite
numbers. Example: 4,6,20,100, etc.
Co-prime numbers: those numbers which do
not have any common factor except 1 are
called co-prime no. example: 15 and 26
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Find nearest perfect square and will divide
the no. by all prime no. till that nearest
square if anyone divides the no. then that no.
is not prime: For ex: take 151
Now the nearest perfect square is (12)²
So divide 151 by 2, 3, 5, 7, 11
Since none of the prime no. completely
divides 151 so that means 151 is a prime no.
A number is said to be divisible by another no. if the remainder
is 0
 A number is divisible by 2 when its unit’s digit is even or 0
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A number is divisible by 3 when the sum of digits is divisible
by 3
A number is divisible by 4 when the number formed by the
last two digits are either 0 or divisible by 4
A number is divisible by 5 when its unit’s digit is 5 or 0
A number is divisible by 6 when it is divisible by 2 and 3
both.
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A number is divisible by 8 when when the number
formed by the last three digits are either 0 or
divisible by 8
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A number is divisible by 9 when the sum of digits
is divisible by 9
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A number is divisible by 10 when its unit digit is 0
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A number is divisible by 11 when the difference
between the sum of the digits in the odd place and
even place is 0 or a multiple of 11.
A number is divisible by 12 when is divisible by 3
and 4 both
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Q. If abc4d is divisible by 4, what is the value of
d?
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Q. A number 344ab5 is divisible by both 9 and
25. Find the number. (Given a + b<8)
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Q. A number 1568X35Y is divisible by 88. What
are the values of X and Y
Q. What is the remainder when 9876532123 is
divisible by 9
a) 1
b) 3
c) 2
d) 4
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Q. which of the following no. is divisible by
99?
a) 32373
b) 37332
c) 32337
d) 23337
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Q. For n, a positive integer greater than 1,
n(n²-1) is always divisible by
a) 6
b) 12
c) 24
d) 48
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Q. What is the remainder when 78X85Y868 is
divisible by 8
a) 1
b) 3
c) 2
d) 4
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Q. If the number 786P86Q is divisible by 8
and 9 both, then values of P and Q are:
a) 4,9
b) 8,6
c) 6,4
d) 6,8
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