Download Aptitude Preparation

Quantitative Aptitude
Preparation
Numbers
Prepared by:
MS. RUPAL PATEL
Assistant Professor
CMPICA, CHARUSAT
Numbers
• Numbers – In Hindu Arabic system, we have
total 10 digits.
• Namely, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• Number is a group of digits called numeral.
• Places of each digit in a numeral
5
6
1
3
0
7
0 9
0
Ten
Crores
Crores
Ten
Lakhs
Lakhs
Ten
Thousand Hundreds Tens
Thousands
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Units
Face value and Place Value
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Types of Numbers
Numbers
Quick Description
Symbol
Natural
{1, 2, 3, ………….}
N
Whole
{0,1, 2, 3, …………}
W
{…..-3, -2, -1, 0, 1, 2, 3, ………}
Z
Integers
Rational
Q
Irrationals Not Rational
Real
All Rational and Irrational
R
Imaginary
I
Complex
C
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Even and Odd Numbers
• The integers which are divisible by 2 are
called even numbers.
• E.g. 0, 2, 4, 6, 8, ……, etc.
• The integers which are not divisible by 2
are called odd numbers.
• E.g. 1, 3, 5, 7, 9, …….., etc.
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Prime Numbers
• A counting number which has only two
factors: 1 and itself is called prime
number.
• E.g. Prime numbers between 1 and 100
• 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,
41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83,
89, 97
• IMP: The only even prime number is 2.
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Composite Numbers and Co-primes
• The natural numbers which are not prime are
called composite numbers.
• E.g. 4, 6, 8, 9, 10, 12, 14, 15, 16, etc.
• Two natural numbers a and b are said to be
co-prime if their HCF is 1.
• Means both a and b has no common factor.
• E.g. (2,3), (4,5), (7,9), (8,11), etc.
• IMP: 1 is neither prime nor a composite
number
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Important
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Modulus or Absolute value of a
Real Number
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Test of Divisibility of Numbers
Div.
Rule
Example (s)
2
If the last digit is an even
number or zero(0)
84, 138, 2, 1920
3
If the sum of the digits
3705
divisible by 3
3+7+0+5 = 15 is divisible by 3
4
If the last two digits
21660, 5100
divisible by 4 or it ends with 21660- since 60 is divisible by 4
‘00’
5100- last 2 digits 00
5
If the last digit is 0 or 5
6
If the number is divisible by
629130
both 2 and 3
last digit 0 so divisible by 2
6+2+9+1+3+0=21 is divisible by 3
8
If the last three digits
81976, 6145000
divisible by 8 or it ends with 81976- since 976 is divisible by 8
‘000’
6145000- last 3 digits 000
865, 1705, 25, 4270, 3300
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Test of Divisibility of Numbers
Div.
Rule
Example (s)
9
If the sum of the digits is divisible
870111
by 9
8+7+0+1+1+1 = 18
10
If the last digit is 0
11
If the difference = sum of digits of
647053
(odd places - even places) is 0 or Odd sum = 6+7+5=18
a multiple of 11
Even Sum = 4+0+3 = 7
Difference = 18 - 7 = 11
7
and
13
730, 20, 5500
If the difference of the number
265216, 2503681
formed by the last 3 digits and 265 – 216 = 49 is divisible by 7
the number formed by the rest 2503 – 618 = 1885
digits divisible by 7 or 13 885 – 1 = 884 is divisible by 13
respectively
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
General Properties of Divisibility
1. If a number A is divisible by another
number B, then any number divisible by
A, will also be divisible by B and all the
factors of B.
• E.g. 84 is divisible by 6
252 is divisible by 84, it will also be divisible
by 6 and also factors of 6, i.e. 2 and 3
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
General Properties of Divisibility
2. If two numbers A and B are divisible by a
number ‘p’, then their sum A+B is also
divisible by ‘p’
• E.g. 225 and 375 are divisible by 5.
• Then, 225 + 375 = 600 is also divisible by 5.
3. If two numbers A and B are divisible by a
number ‘p’, then their difference A-B is also
divisible by ‘p’
• E.g. 126 and 507 are divisible by 3.
• Then, 507 - 126 = 381 is also divisible by 3.
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Tables of 1 to 20
• Must Memorize
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Squares of 1 to 100
• Must Memorize squares of 1 to 30
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Test of Prime Numbers
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Test of Prime Numbers
• E.g.
I. 137
122 = 144 > 137
prime numbers < 12 are 2, 3, 5, 7, 11
None of them divides 137. Thus it is a prime.
II. 203
152 = 225 > 203
Prime numbers < 15 are 2, 3, 5, 7, 11, 13
203 is divisible by 7. Thus, it is not a prime.
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Divisibility and Remainder
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Methods to find Completely
Divisible by another number
• Least number that can be subtracted to
make it exactly divisible
• Answer = Remainder
• Least number that can be added to
make it exactly divisible
• Answer = Divisor - Remainder
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
• E.g. Find the least number, that must
subtracted from and added to a given number
5029, to make it exactly divisible by 17. Also
find the numbers in each case.
• Soln
17) 5029 (295
34
162
153
99
85
14
Least number subtracted = remainder = 14
i.e. 5029 – 14 = 5015
Least number added = divisor – remainder = 17 – 14 = 3
i.e. 5029 + 3 = 5032
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Greatest n-digit and Least n-digit
number exactly divisible by a number
• Greatest
n-digit
number
exactly
divisible by a divisor ‘d’
• Answer = Greatest n-digit no. – Remainder
• Least n-digit number exactly divisible
by a divisor ‘d’
• Answer = Least n-digit No. + (Divisor –
Remainder)
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
• E.g. Find the
(a) Greatest 3-digit number divisible by 13
(b) Least 3-digit number divisible by 13
• Soln
13) 999 (76
13) 100 (7
91
91
89
9
78
11
Greatest 3-digit number = 999 – 11 = 988
Least 3-digit number = 100 + (13 – 9) = 104
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Remainder Rules
No.
Type of Problem
Rule
1
When a number is divided by x
then leaves remainder R, when
same number is divided by y,
what will be the remainder?
2
If two different numbers a and b R = (r1+r2) – divisor
are divided by the same divisor
leaves remainder r1 and r2, If R becomes negative, then
respectively, then their sum (a+b) R = r1+r2
if divided by same divisor will
leave remainder R
3
When two numbers after being 1. Find the difference of
divided by the same divisor leave
given two numbers
the same remainder, then the 2. Find the factors of it.
difference of those two numbers 3. Find the divisor according
must be exactly divisible by the
to n-digits asked
same divisor.
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Remainder Rules
No.
Type of Problem
Rule
4
If a given number is divided True Remainder
successively by the different = (r1) + (r2 x d1) +(r3 x d1 x r2)
factors of the divisor leaving
remainders r1, r2 and r3,
respectively, then the true
remainder is
5
When (x + 1)n is divided by x, Remainder is always 1.
then remainder is
When x and n are naturals
6
When (x – 1)n is divided by x, When n = even no, R = 1,
then remainder is
When n = odd no, R = (x – 1)
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Sum Rules on Natural Numbers
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Algebraic Formulas
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Algebraic Formulas
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Last digit (Unit’s digit) of (xyz)n
• The given number is (xyz)n then last digit of a
number is z. Based on this z, we, decide the last
digit of a number when its power is calculated.
Value of z
Divide n
0, 1, 5, or 6
-
4 or 9
By 2
2, 3, 7 or 8
By 4
Value of
Remainder
1
0
1
2
3
4
Last digit of zn
0, 1, 5, or 6,
respectively
z
z2
z
z2
z3
z4
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Number of zeroes at the End of Product
• There are 2 reasons of 0’s at the end of product
1. If there is any 0 at the end of the number.
E.g. 7 x 20 = 140
2. If 5 or multiple of 5 multiplied by any even no.
E.g. 45 x 12 = 540
Steps
1. Find the factors of given number
2. Count number of 2’s and 5’s
3. Number of 0’s = Min (No. of 2’s and No. of 5’s)
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
• E.g. Find the number of zeroes at the
end of the product of
15 x 32 x 25 x 22 x 40 x 75 x 98 x 112 x 125
= (5x3) x (25) x (52) x (11x2) x (23x5) x (52x3)
x (2x72) x (24x7) x (53)
No. of 2’s = (25+1+3+1+4) = 214 = 14
No. of 5’s = (51+2+1+2+3) = 59 = 9
Min(9,14) = 9
Hence, There are total 9 zeroes at the end of
the product.
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Practice sums
1. The difference between the local value
and face value of 6 in the numeral
33186890 is
(a) 6888
(b) 6886 (c) 5940 (d) None
2. The difference between place value’s of
two 4 in the numeral 32645149 is
(a) 0
(b) 45130 (c) 39960 (d) None
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Practice sums
3. Find the least value of * for which 7*5462
is divisible by 9.
(a) 2
(b) 0
(c) 3
(d) 5
4. Find the least value of * for which
4832*18 is divisible by 11.
(c) 7
(a) 1
(b) 5
(d) 9
5. Show that 52563744 is divisible by 24.
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Practice sums
6. What least number must be subtracted
from 1672 to obtain a number which is
completely divisible by 17.
(c) 6
(a) 3
(b) 4
(d) 5
7. What least number must be added to
2010 to obtain a number which is
completely divisible by 17.
(b) 4
(a) 2
(c) 7
(d) 9
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Practice sums
8. On dividing 12401 by a certain number,
we get 76 as quotient and 13 as
remainder, What is the divisor?
(a) 162
(b) 165 (c) 163 (d) 161
9. On dividing a certain number by 342, we
get 47 as remainder. If the same number
is divided by 18, what will be the
remainder?
(d) 11
(a) 1
(b) 3
(c) 5
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Practice sums
10. What is the unit digit in the product
(684 x 759 x 413 x 676)
(b) 8
(a) 4
(c) 1
(d) 3
11.What is the unit digit in the product
(3547)153 x (251)72?
(a) 1
(b) 2
(c) 4
(d) 7
12. What is the unit digit in
(264)102 + (264)103
(a) 4
(c) 3
(d) 1
(b) 0
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Practice sums
13. Find the total number of prime factors in
the product {(4)11 x (7)5 x (11)2} ?
(a) 22
(b) 25
(c) 29
(d) 27
14. A number when successively divided by
3, 5, and 8 leaves remainders 1, 4, and 7
respectively.
Find
the
respective
remainders if the order of divisors be
reversed.
(a) 4,2,6 (b) 6,4,2 (c) 2,4,6 (d) 6,2,4
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Practice sums
15. Find the remainder when 231 is divided by 5.
(a) 2
(b) 3
(c) 9
(d) 7
16. How many natural numbers between 17 and
80 are divisible by 6?
(c) 11
(a) 12
(b) 16
(d) 14
17. (6 + 15 + 24 + 33 + …….. + 105) = ?
(b) 667 (c) 678 (d) 677
(a) 666
18. Find the sum of all even natural numbers
less than 75.
(a) 1407 (b) 1466 (c) 1403 (d) 1406
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Practice sums
19. What is the unit digit in 7105
(c) 7
(a) 1
(b) 5
(d) 9
20. What is the unit digit in the product
(365 x 659 x 771)?
(c) 4
(a) 1
(b) 2
(d) 6
21. What is the unit digit in {(6374)1793 x
(625)317 x (341)491?
(b) 2
(c) 3
(d) 5
(a) 0
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Practice sums
22. 587 x 999
(a) 586413
(c) 614823
23. 1904 x 1904
(a) 3654316
(c) 3625216
24. 107 x 107 + 93 x 93
(a) 19578
(c) 20098
(b) 587523
(d) 615173
(b) 3632646
(d) 3623436
(b) 19418
(d) 21908
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Practice sums
25.Which one of following numbers is
exactly divisible by 11?
(a) 235641
(b) 245642
(c) 315624
(d) 415624
26. The sum of first 45 natural numbers is
(a) 1035
(b) 1280
(c) 2070
(d) 2140
27. The sum of even numbers between 1
and 31 is
(a) 6 (b) 128 (c) 240 (d) 512
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Practice sums
28.What smallest number should be added to
4456 so that it is completely divisible by 6?
(a) 4 (b) 3
(c) 2
(d) 1415624
29. The largest 4 digit number exactly
divisible by 48 is
(b) 9768 (c) 9988 (d) 8888
(a) 9944
30. The difference of two numbers is 1365.
On dividing the larger number by the
smaller, we get 6 as quotient and 15 as
remainder. What is the smaller number?
(b) 270 (c) 295 (d) 360
(a) 240
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa