Session 9: Identifying the Last Digit | A NEW OUTLOOK – CSIR-NET Series

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Live Lecture

Theory

Quizzes

Last Digit of Power $121^{81}$

That’s right!

That’s wrong! Hint: Turns out 121 ends with 1 – then digit in its units place has to be…

Last Digit of Power $2017^{2017}$ is

That’s wrong! Hint: Exponent is 2017
Units digit of base is 7
Cyclicity of 7 is 4
Reminder when 2017 is divided by 4 is 1.. Thus…

That’s right!

That’s wrong! Hint: Exponent is 2017
Units digit of base is 7
Cyclicity of 7 is 4
Reminder when 2017 is divided by 4 is 1.. Thus…

Last Digit of Power $26591749^{110016}$ is

That’s right!

That’s wrong! Hint: Exponent is 110016
Units digit of base is 9
Cyclicity of 9 is 2
Reminder when 110016 is divided by 2 is 0
Thus…

The digit at the unit’s place of $489^{86} - 351^{63}$ is

That’s wrong! Last digit of $489^{86}$ (Cyclicity 2) is 1.
Last digit of $351^{63}$ (Cyclicity 1) is 1.
Thus the last digit is…

That’s right!

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Session 8: Perfect Squares | A NEW OUTLOOK – CSIR-NET Series

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Live Lecture

Warm Up Questions

Last Digit of 71 + 72 + 73 + 74 + 75 + 76

That’s right!

That’s wrong!

Last Digit of 70 X 71 x 72 x 73 x 74 x 75

That’s right!

That’s wrong! Hint: It’s 70 x….

Last Digit of 71 x 72 x 73 x 74 x 75 x 76

That’s right!

That’s wrong! Hint: It’s 72 x 75 x….

Definitions

Perfect Square is a natural number that can be written as the product of two equal factors.,

For example: 4, 25, 81, 441, …

Observations

Last Digit of the Multiple is influenced only by the Last Digits of the Numbers.
No perfect square ends with 2, 3, 7, 8.
No perfect square ends with an odd number of zeros.
Upon prime factorization, all their prime factors have even multiplicities.

Which of the following numbers is a perfect square? (CSIR-NET NOV 2020)

48841

That’s right!

58287

That’s wrong! Hint: Check the observations above!

68763

That’s wrong! Hint: Check the observations above!

38262

That’s wrong! Hint: Check the observations above!

Which of the following numbers is a perfect square? (CSIR-NET DEC 2014)

1022121

That’s right!

2042122

That’s wrong! Hint: No perfect square ends with…

3063126

That’s wrong! Hint: For a perfect square, when units place is 6, tens place will be…

4083128

That’s wrong! Hint: No perfect square ends with…

Which of the following 7-digit numbers CANNOT be perfect squares? (CSIR-NET DEC 2019)
A = 45xyz26, B = 2xyz175, C = xyz3310

Only A

That’s wrong! Hint: Turns out C ends with 5 – then digit in its tens place has to be…

Only B

That’s wrong! Hint: Turns out C ends with 0 – then digit in its tens place has to be…

Only C

That’s wrong! Hint: Turns out A ends with 6 – then digit in its tens place has to be…

All three

That’s right! All three cannot be perfect squares!

Question

Abhirup had a question on “Upon prime factorization, all their prime factors have even multiplicities.” How does that work?

Here’s how. For example, consider 36. 36 = 6² .

Prime factorisation of 36 is given by 36 = 2² x 3².

Another example will be 484 = 22² = 11² x 2².

Why is it so? We can give prime factorisation of any natural number, say n = a^m x bⁿ x c^p x …, where a, b, c, … are prime numbers.

When you square it, n² = (a^m)² x (bⁿ)² x (c^p)² x … = a^2m x b²ⁿ x c^2p x …

Thus, all the prime numbers have even multiplicities.

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Perfect Squares and More

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Session 7: Surds and Exponents | A NEW OUTLOOK – CSIR-NET Series

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Exponent Form

When a number is multiplied by itself several times, it can be written in a short form with the help of exponents. Example: 5×5×5×5=5⁴. Here, 5⁴ is called an exponential expression.

The number multiplied by itself, again and again, is the base. The number of times the number appears or is multiplied is the exponent. In 5⁴, 5 is the base and 4 is the exponent.

Surds

The values in the square root or cube root or any other roots, which cannot be further simplified into whole numbers or integers, are known as a surd. Example: The value of √5.

Laws of Exponent

a^m× aⁿ = a^m+n
a^m/ aⁿ= a^m-n
a^m× b^m= (ab)^m
a^m/ b^m= (a/b)^m
(a^m)ⁿ = a^mn
${a^m}^n=a^{\left(m^n\right)}$
a^(1/m) = ^m√a
a^-m = 1/a^m
a^m/n = ⁿ√a^m
a⁰ = 1

Which of the following values is same as ${{2^2}^2}^2$ ? (CSIR-NET FEB 2022, JUNE 2017)

$2^6$

That’s wrong! Hint: $a^m^n=a^{\left(m^n\right)}$

$2^8$

That’s wrong! Hint: $a^m^n=a^{\left(m^n\right)}$

$2^{16}$

That’s right!

$2^{222}$

That’s wrong! Hint: $a^m^n=a^{\left(m^n\right)}$

$4^0+4^2+\frac{1}{4^2}+4^\frac{1}{2}+\frac{1}{4^\frac{1}{2}}$ equals (CSIR-NET December 2012)

$4^0$

That’s wrong! Hint: Use the laws of exponents!

$4^{2\frac{1}{2}}+4^{-2\frac{1}{2}}$

That’s wrong! Hint: Use the laws of exponents!

$19\frac{9}{16}$

That’s right!

$22\frac{9}{16}$

That’s wrong! Hint: Use the laws of exponents!

Some Important Expansions

(a + b)² = a² + 2ab + b²
(a – b)² = a² – 2ab + b²
a² – b²= (a + b)(a – b)
(a + b)³ = a³ + 3a²b + 3ab² + b³
(a – b)³ = a³ – 3a²b + 3ab² – b³

Suppose $729^{22\over3} 81^{x\over2} = 3$ . The value of x is (CSIR-NET JUNE 2023)

-21

That’s wrong! Hint: 729 = 9 x 9 x 9, 81 = 9 x 9!

-21.5

That’s right!

-22

That’s wrong! Hint: 729 = 9 x 9 x 9, 81 = 9 x 9!

-20.5

That’s wrong! Hint: 729 = 9 x 9 x 9, 81 = 9 x 9!

Which of the following is the largest? (CSIR-NET JUNE 2019)

$2^{50}$

That’s wrong! Hint: Try making the exponent same!

$3^{40}$

That’s right!

$4^{30}$

That’s wrong! Hint: Try making the exponent same!

$5^{20}$

That’s wrong! Hint: Try making the exponent same!

Which of the following is the largest? (CSIR-NET JUNE 2017)

$2^{500}$

That’s wrong! Hint: Try making the base same!

$3^{400}$

That’s right!

$4^{300}$

That’s wrong! Hint: Try making the exponent same!

$5^{200}$

That’s wrong! Hint: Try making the exponent same!

Which of the following numbers is the largest? $2^{3^4}, 2^{4^3}, 3^{2^4}, 3^{4^2}, 4^{2^3}, 4^{3^2}$ (CSIR-NET DEC 2012)

$2^{3^4}$

That’s right!

$3^{4^2}$

That’s wrong! Hint: Try making the base same!

$4^{3^2}$

That’s wrong! Hint: Try making the base same!

$4^{2^3}$

That’s wrong! Hint: Try making the base same!

Here’s a comic to wind up the discussion:

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Session 6: Number Systems, Digits and Place Values | A NEW OUTLOOK – CSIR-NET Series

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Live Lecture

Number Systems

Place value

Place value is the value of each digit in a number. The value of every digit in a number is different based on its position. A number might have two similar digits but different values, which is decided by the position that the digit holds in the number.

Place Value and Face Value

Given the number is 5369.

Face value of 9 is 9 and Place value of 9 is 9
Face value of 6 is 6 and place value of 6 is 60 (Sixty)
Face value of 3 is 3 and place value of 3 is 300 (Three hundred)
Face value of 5 is 5 and place value of 5 is 5000 (Five thousand)
Therefore, we can see, the face value of the digits in number 5369 represents the actual value of the digit.

Representing numbers in Decimal Number System

2-digit number – 10x+y

3-digit number – 100x+10y+z

…

In general, a n-digit number $a_1 a_2 \cdots a_n = 10^{n-1}a_1 + 10^{n-2}a_2 + \cdots + 10^{1}a_{n-1} + 10^0 a^n$

Representing numbers in any Number System

Watch the video to learn more.

Here’s the flashcard:

Conversion of Number Systems

Refer to the lecture for details.

0-40 in Different Number Systems (Flash Card)

Buy/refer to the flashcard series for more content like this. Click here for full catalogue.

Convert $67_8$ to other number systems.

Hexadecimal: Show Answer

Decimal: Show Answer

Binary: Show Answer

Convert $10_{10}$ to other number systems.

Hexadecimal: Show Answer

Octal: Show Answer

Binary: Show Answer

1010

If 25 is written as
$11001(1\times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 0\times 2^1 +1\times 2^0)$
Then how will 101 be written? (CSIR-NET SEPT 2022 – CS)

1010101

That’s wrong! Is it a familiar number system?

1100111

That’s wrong! Is it a familiar number system?

1101101

That’s wrong! Is it a familiar number system?

1100101

Here, 25 is converted to binary form.

Consider a number 54 expressed in a base different from 10. What is the base of this number system if its equivalent value in the decimal system is 49? (CSIR-NET JUNE 2018)

That’s wrong! If x is the base, then $49=5x^1+4x^0$ … Then x is?

That’s right!

N is a two-digit number such that the product of its digits when added to their sum = N. The unit digit of N would be: (CSIR-NET JUNE 2017)

That’s wrong! Hint: Write N = 10x + y. Then Product of digits = xy and the Sum of digits = x + y.

That’s right!

Choose the four-digit number, in which the product of the first & fourth digits is 40 and the product of the middle digits is 28. The thousands digit is as much less than the unit digit as the hundreds digit is less than tens digit. (CSIR-NET JUNE 2016)

5478

That’s right!

5748

That’s wrong! Hint: The thousands digit is < Unit digit and hundreds digit is less than tens!

8745

That's wrong! Hint: The thousands digit is < Unit digit and hundreds digit is less than tens!

8475

That's wrong! Hint: The thousands digit is < Unit digit and hundreds digit is less than tens!

If D = ABC + BCA + CAB where A, B, C are decimal digits, then D is divisible by? (CSIR-NET DEC 2018)

37 and 29

That's wrong! Hint: Try taking the simplest example!

37 but not 29

That's right!

29 but not 37

That's wrong! Hint: Try taking the simplest example!

Neither 29 nor 37

That's wrong! Hint: Try taking the simplest example!

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Session 5: Divisibility | A NEW OUTLOOK – CSIR-NET Series

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Live Lecture

Tests for Divisibility

Divisibility by 2: Check if the last digit is even. i.e., 0, 2, 4, 6 or 8.
Divisibility by 3: Check if the sum of its digits is divisible by 3.
Divisibility by 4: Check if the number formed by the last 2 digits is divisible by 4.
Divisibility by 5: Check if its last digit is 0 or 5.
Divisibility by 6: Check if the number is divisible by both 2 and 3.
Divisibility by 7: To check whether a given number is divisible by 7, subtract twice the unit digit from the number represented by the remaining digits of the number to obtain an integer. The original number is divisible by 7 if the obtained integer is divisible by 7.
Divisibility by 8: Check if the number formed by the last 3 digits is divisible by 8.
Divisibility by 9: Check if the sum of the digits is divisible by 9.
Divisibility by 11: Mystery! It’s hidden somewhere in this page! Try to find it!

Division Algorithm

Dividend = Divisor × Quotient + Remainder

For example: 62 = 10 × 6 + 2 and 60 = 10 × 6 + 0

Factors and Factorisation

Prime factorisation is a process of factoring a number in terms of prime numbers i.e. The factors will be prime numbers.

The simplest algorithm to find the prime factors of a number is to keep on dividing the original number by prime factors until we get the remainder equal to 1.

Factors: A number is a factor of another, if the former exactly divides the latter without leaving a remainder (remainder is zero). For example: 5 is a factor of 30.

Multiples: A multiple is a number which is exactly divisible by another. For example: 30 is a multiple of 5.

Greatest Common Divisor (HCF)

The GCD or HCF of two or more than two numbers is the greatest number that divides each of them exactly.

Least Common Multiple (LCM)

The LCM of two or more numbers is the smallest number which is exactly divisible by each one of the given numbers.

Refer to the lecture for methods to find GDC/LCM and important properties of it!

GCD of 12, 24, 15 is

That’s wrong, try again!

That’s right!

That’s wrong, try again!

In a ten-digit mobile number 999ABCDEEE, A, B, C and D are distinct prime numbers. The mobile number is never divisible by (CSIR-NET NOV 2020)

That’s wrong, try again!

That’s right!

That’s wrong, try again!

Which is the odd one out based on a divisibility test?
154, 286, 363, 474, 572, 682 (CSIR-NET JUNE 2017)

474

That’s right!

572

That’s wrong! Hint: Divisibility test for 11: A number is divisible by 11 if the difference of the sum of the digits in the even and odd positions in the number is divisible by 11.

682

That’s wrong! Hint: Divisibility test for 11: A number is divisible by 11 if the difference of the sum of the digits in the even and odd positions in the number is divisible by 11.

154

That’s wrong! Hint: Divisibility test for 11: A number is divisible by 11 if the difference of the sum of the digits in the even and odd positions in the number is divisible by 11.

How many integers in the set {1,2, 3, …, 100} have an odd number of divisors? (CSIR-NET Feb 2022)

That’s right!

That’s wrong! Hint: An integer has exactly 3 divisors if the integer is the square of a prime number as the three factors are 1, the integer itself and its square root.

More Materials

Watch this video from previous lecture for more Previous Year Question and discussion:

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Someone had enquired about mathematics sessions. For mathematics related content, please check out our sister project, Mathematicos.in and the YouTube Channel.

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Session 4: Types of Numbers | A NEW OUTLOOK – CSIR-NET Series

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Live Lecture

Session Outline

A lot of familiar (or less familiar) words about numbers.
Exploring their precise meaning.
Building up for problems to come in upcoming sessions.
Why?
- Questions may contain these words.
- Some minor questions we never asked
  - Is 0 a Natural number?
  - Is 19.333333… a rational number?
  - Is π=22/7?

Interesting Sets of Numbers We Talked About

“Precise” Meanings

Natural Numbers: The counting numbers like 1, 2, 3, 4… are called natural numbers.

Whole numbers: Natural numbers along with zero are called whole numbers.

Integers: All counting numbers including zero and the negatives of the counting numbers form the set of integers. Hence …, –3, –2, –1, 0, 1, 2, 3… are all integers.

A few terms to remember: “Positive Integers”, “Negative Integers”, “Non-negative integers” and “Non-positive integers”.

Classification of Integers: Even (divisible by 2), Odd (not divisible by 2).

Rational Numbers (Fractions): A rational number represents a part of a whole or, more generally, any number of equal parts.

Mixed Fraction: An improper fraction can be expressed as a whole number and a proper fraction. This expression is called a mixed fraction.

Decimal representation of rational numbers: When a rational number is written in decimal points, the digits after the decimal will

Either stop after a while (Eg. 1.343443)
Or repeat the same pattern forever (Eg. 1.546754675467…. “5467” repeats forever).

Irrationals: Numbers with infinite decimal expansion without any pattern are called irrationals. π is an irrational number.

Real numbers: Rational numbers, along with irrational numbers are called real numbers.

Quiz

Which is the smallest natural number given in the options?”

Remember? Natural numbers are counting numbers… But zero is not a natural number!

That’s right

-1

Natural numbers are Positive Integers!

Try again!

True or False: We have a rational number which is an irrational number.

TRUE

An irrational number is a number which is not rational. Hence there cannot a rational number which is also irrational!

FALSE

That’s right

True or False: Pi = 22/7

TRUE

Try again!

FALSE

Pi is an irrational number while 22/7 is rational. In fact, 22/7 is a rational approximation of pi.

Additionally, we used terms like, “Sets”, “Venn Diagram”, “[set] contained in [another set]” etc.

Observations from Chair-Arranging Experiment

Odd number of chairs cannot be arranged in two rows to make a perfect rectangle.
Even number of chairs can be arranged in two rows to make a perfect rectangle.
Addition
- Even + Even = Even
- Odd + Odd = Even
- Even + Odd = Odd
- Odd + Even = Odd
Multiplication
- Even × Even = Even
- Even × Odd = Even
- Odd × Even = Even
- Odd × Odd = Odd

Classification of Natural Numbers

Prime Numbers: A natural number which has exactly two positive factors (divisors), namely itself and 1. For example, 2, 3, 5, 7, 11, …

Composite Numbers: A natural number which has more than two positive factors. For example: 4, 6, 8, 9, 10,…

Note: 1 is neither prime nor composite as it has only one factor. Note the correction from how it was mentioned in the video lecture.

Quizzes

Ture/False: Is 0 a prime number?

TRUE

0 is not a natural number, hence not prime!

FALSE

True. 0 is not a natural number to begin with!

Why is 1 not a prime?

See answer

1 is a natural number but it has only one positive divisor, 1. Hence it cannot be prime.

How many prime numbers are there less than 10?

That’s wrong!

That’s right. 2,3,5,7

That’s wrong!

How many even prime numbers are there less than 1000?

That’s wrong! Try again!

True! Any even positive integer will be divisible by itself, 1 and 2 (by the definition of even numbers).

That’s wrong! Try again!

Why is composite numbers not called “natural numbers with more than two divisors”?

Show explanation

Then 1 (which has only one positive integer divisor) will not be a composite or prime number.

Here’s a cartoon to end it in a fun note:

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Session 3: Tips and Tricks | A NEW OUTLOOK – CSIR-NET Series

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Live Class

Tips

Tip 1: Have an Objective Approach

When you see a question, do the following:

Read the question and the options given
Choose the less time-consuming method
Solve the question

Six persons P,Q,R,S,T and U sit around a circular table with equal distance between neighbours. P is to the immediate left of R. T and S do not sit next to each other, and T and R are diametrically opposite to each other. Which of the following is NOT possible? (CSIR NET FEB 2022)

Q and P are sitting diametrically opposite to each other

Q and P are sitting diametrically opposite to each other: This is possible. We can place Q and P in two opposite seats.

P and U are sitting diametrically opposite to each other

P and U are sitting diametrically opposite to each other: This is possible. We can place P and U in two opposite seats.

S and T are sitting diametrically opposite to each other

S and T are sitting diametrically opposite to each other: This is not possible because T and R are already diametrically opposite to each other.

S and Q are sitting diametrically opposite to each other

S and Q are sitting diametrically opposite to each other: This is possible. We can place S and Q in two opposite seats.

Tip 2: Time factor

2 minutes – Maximum Time for any question
30 minutes – Maximum Time for Part A Section
Assign Time for Part A, Part B and Part C

Tip 3: Choose the Right Method

Practice makes it perfect! Train yourself to choose the right method to solve the question in the shortest time. Read Session 2 Notes or Buy the Book for more details.

A and B complete a work in 30 days. B and C complete the same work in 24 days whereas C and A complete the same work in 28 days. Based on these statements which of the following conclusions is correct? (CSIR-NET NOV 2020)

C is the most efficient and B is the least efficient.

Try again!

B is the most efficient but the least efficient one cannot be determined.

Try again!

C is the most efficient but, the least efficient one cannot be determined.

Try again!

C is the most efficient and A is the least efficient.

Time taken by C to complete the work = 45.4 days
Time taken by A to complete the work = 73 days
Time taken by B to complete the work = 50.9 days

Tip 4: Choose the Right Question

Approximate Question Distribution

Easy questions: 7 to 8
Medium level questions: 5 to 6
Difficult questions: 7 to 8

Train yourself to identify the difficulty of the question in the shortest time!

There are three fathers and six sons in a room. If every father’s son(s) and every son’s father are present, then the minimum number of individuals in the room are (CSIR-NET FEB 2022)

Try again!

That’s right! A is the grandfather of four children of B and C, who is are the children of A.

Try again!

Try again! This might be the natural option, but the question is bound to be tricky!

Tip 5: Finding the Catch in the Question

Find the Word or Phrase which gives you the solution at the earliest!

What is the volume of soil in an open pit of size 2 m × 2 m × 10 cm? (CSIR-NET FEB 2022, DEC 2017)

4 metre cube

Try again!

0.4 metre cube

Try again!

0 metre cube

Whatever be the size, this pit contains nothing.

40 metre cube

Try again!

Tip 6: Ignore Irrelevant Data

In a population of 900, the number of married couples is as much as the number of singles. There are 100 twins of which 50 twins are singles. The population has 400 females in all. What is the number of married persons? (CSIR-NET DEC 2019)

325

Try again!

600

That’s right

250

Try again!

300

Try again!

Tip 7: Choose the Right Starting Point

Seven persons A, B, C, D, E, F, and G are sitting in a row. E and B are sitting adjacent to each other. F is sitting between D and G. If C is sitting four places left of F, who among the following cannot be sitting at the centre? (CSIR-NET NOV 2020)

Try again!

That’s right

Other Tips

8. Be Fast in Arithmetic

9. Learn Basic Formula

10. Double check the keywords in the question

11. Find the Easy “Common Sense” Questions

12. Beware of very simple answers

13. Consider all the cases of the question

14. The power of Negative Thinking!!! Don’t give in to it!

Watch the video for more details!

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Session 2: Shortcut Methods | A NEW OUTLOOK – CSIR-NET Series

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Live Class

Tricks Discussed and More

You can learn more of such tricks with examples and practice questions from our book!

Questions

Ajay, Bunty, Chinu and Deb were agent, baker, compounder and designer, but not necessarily in that order. Deb told baker that Chinu is on his way. Ajay is sitting across the designer and next to the compounder. The designer didn’t say anything. What is each person’s occupation? (CSIR-NET DEC 2014)

Ajay-Compounder, Bunty-Designer, Chinu-Baker, Deb-Agent