Numbers Session 1 Class 1 Slides

1. Session for CAT
Alosies George
IIM Calcutta
director@georgeprep.com
+91- 9985-372-371
Numbers Session 1 Class 1
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2. Session for CAT
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Session 1
Classification and rational, irrational
numbers
Recurring decimals
Prime numbers and composite
numbers
Properties of Squares and Cubes
Divisibility rules
Osculators for 7, 13, 17 and the like
Modulo arithmetic understanding of
divisibility rules
Divisibility Rules of Numbers in
Different Bases
Session 2
Odd-Even Rules
Counting concepts
HCF and LCM
HCF and LCM with remainders
Successive division
Factors of a number and other related
formula with reasons behind the
formula
Session 3
Units digits
Involution
Remainders –Basics
Pattern method and Euler’s Totient
Theorem
Application of Chinese, Wilson’s,
Fermat’s, Carmichael etc. in
Remainders
Miscellaneous - Fibonacci series
Miscellaneous - Pigeon hole principle

3. Session for CAT
Classification
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Complex
Numbers (a+ib)
Imaginary
Numbers
Real
Numbers
Irrational
Numbers
Rational
Numbers
Integers
Negative Integers Whole Numbers
Zero
Natural
Numbers
Fractions
Proper
Fraction
Improper
Fraction

4. Session for CAT
Rational and Irrational Numbers
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Rational Numbers Irrational Numbers
Numbers that can be expressed in
the form p/q, where p and q are
integers and q ≠ 0
Numbers that cannot be expressed in the
form p/q, where p and q are integers and q
≠ 0
(or)
Non terminating and non repeating
decimals
Examples:
π(pi) = 3.141592653589…..
e(Euler’s number) = 2.7182818284590….
Φ(Golden Ratio) = 1.6180339887…..
√ not perfect squares

5. Session for CAT
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Interactions between Rational Numbers and Irrational
Numbers
Between two rational numbers there is an irrational
number
Sum
Difference
Product

6. Session for CAT
Recurring Decimals
Pure Recurring Decimals Mixed Recurring Decimals
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7. Session for CAT
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Ans: Option 4
R1, R2 R3 and R4 are four recurring decimals given by
R1 = 0. p1 𝑝2 𝑝3 𝑝4
R2 = 0. p1 p2 p3 𝑝4
R3 = 0. p1 p2 p3 p4
R4 = 0. 𝑝1 𝑝2 𝑝3 𝑝4
Where p1 , p2 ,p3 and p4 are single digit natural numbers.
Which of the following numbers when multiplied by at least one of the
above mentioned decimals will result in an integer irrespective of the
values of p1,p2,p3 and p4?
1.19900
2.9980
3.1800
4.19980

8. Session for CAT
Prime Numbers and Composite Numbers
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Prime Numbers Composite Numbers

9. Session for CAT
Properties of Prime Numbers
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- Exactly two factors
- Digital sum = 1,2,4,5,7 or 8( except for 3)
- Number of prime numbers less than 100
25
9-6-6-4
442-232-232-1
- Let P be a prime number >3, then
P can be expressed in one of the forms 6k+1 or 6k+5 Why??

10. Session for CAT
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Primality Test – Is that Prime?
Is 271 prime?

11. Session for CAT
- CAT 2003 Retest
Answer: Option 4
Primality Test – Is that Prime?
Let n (>1) be a composite integer such that √n is not an integer.
Consider the following statements:
A.n has a perfect integer-valued divisor which is greater than 1 and
less than √n
B.n has a perfect integer-valued divisor which is greater than √n but
less than n
1.Both A and B are false
2.A is true but B is false
3.A is false but B is true
4.Both A and B are true
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12. Session for CAT
HCF =1
Any two consecutive natural numbers are co primes
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Co-Primes

13. Session for CAT
Natural number P is called a perfect number if sum of all its factors
= 2(P)
Examples : 6,28,496
Trivia : Nobody has been able to find any odd Perfect Number yet!
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Perfect Numbers

14. Session for CAT
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-1 0 +1
- Units digits = 0-1,4-5-6,9
- Digital sum = 1-4-9-7
- Difference between the squares of two consecutive natural
numbers. How it helps to find Square of a number?
- Remainders when squares are divided by 3 and 4 ( and proof)
Change of values on squaring a number
Properties of Squares

15. Session for CAT
How many perfect squares can be formed using 1,3,4,6
and 9 exactly once in the number?
Properties of Squares
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Answer: 0

16. Session for CAT
- Units digits = all the digits are possible
- Digital sum = 1, 8 and 9
Change of values on cubing a number
-1 0 +1
Properties of Cubes
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17. Session for CAT
One day the headmaster of a school assembled 500 students studying in
class 5 or class 4 and started distributing chocolates. He gave 11 chocolates
to the first student, 15 to the second student, 19 to the third student and so
on till the 500th student. He instructed the students to go to class 1, 2 and 3
and distribute the chocolates such that the number of chocolates each of
them distributes to a student should be equal to the number of students to
whom they distribute.
How many of the student of class 5 or 4 will not be able to distribute their
chocolates?
1. 500
2. 250
3. 10
4. None of these
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Answer: Option 1