Last two digit
Last two digits

I am dividing this method into four parts and we will discuss each part one by one:

a. Last two digits of numbers which end in one
b. Last two digits of numbers which end in 3, 7 and 9
c. Last two digits of numbers which end in 2
d. Last two digits of numbers which end in 4, 6 and 8

Before we start, let me mention binomial theorem in brief as we will need it for our calculations.

Last two digit
Binomial theorem

Last two digits of numbers ending in 1

Let’s start with an example.

What are the last two digits of 31786?

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last non zero digit

At every step in life, we face diverse problems. They may be related to our family, health, education, career and much more. There can be no one who says that I never had a problem. Srimad Bhagwatam [10.14.58] says ‘padam padam yad vipadam’ which means that at every step in life we face problems. Now how to eradicate our problems is the biggest problem. One thing is sure that we cannot escape the discomfort created because of the problems of life. So the wiser advice will be to find out the root cause of our problems and try to kill it. And remember, we must help each other by solving each other’s problems. That makes life smoother. You get dependable and caring friends which you’d like to carry along throughout life.

Coming back to our math worries, many students have repeatedly asked us to create a Quant corner where we can post our (beautiful) solutions to beautiful math problems and discuss some small concepts. So here we are! This place is open for discussions as that is the best way of learning. So let’s start today’s mathematical extravaganza.

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Grammar

  1. A NUMBER VS THE NUMBER

As a stand-alone word and as a collective noun, “number” can take a singular or a plural form.

e.g. Two hundred persons were at the party; the number(s) is (are) just astounding.

However, when preceded by an article and followed by the preposition “of”, “number” is singular and the verb that follows “number” will be conjugated singular or plural depending on whether there is a definite or indefinite article in front. The expression ‘the number of . . .’ is singular, while ‘a number of . . .’ is plural.

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Grammar
Grammar

. If a Singular object is separated by a comma from an accompanying phrase it remains singular.

Correct: Ricky, accompanied by his wife was at the party.

Incorrect : Rickey, accompanied by his wife were at the party.

. A relative pronoun should be placed as close to its antecedent as possible.

Correct : I have read writings of Plato, who was a disciple of Socrates.

Incorrect : I have read Plato’s writing who was a disciple of Socrates.

Antecedent- The noun for which the pronoun is used.

Plato is the noun and its antecedent is who (relative pronoun)

. A reflexive pronoun cannot be used alone as a subject of the verb. It must be preceded by some other noun or pronoun.

Correct : Her mother and I were standing there.

Incorrect : Myself and her mother were standing there.

Reflexive pronoun- Formed by adding self/selves to the pronoun.

  1. An intransitive verb will always be followed by a subject pronoun.

Correct : It must have been she at the door.

Incorrect : It must have been her at the door.

Correct : I wish I were he.

Incorrect : I wish I were him.

Correct : He didn’t know that it was I who did it.

Incorrect : He didn’t know that it was me who did it.

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Average mixture
Average mixture

 

The concept of averages is hardly a new concept at all. If asked, all of you would give me the following formula for calculating

Average mixture
Average formula

 

 

 

So far so good. But if I ask all of you to solve a simple problem many of you would reach for their pens.

The average score of three students A, B, and C is 50. When the score of another student D is added to the group, the average score becomes 47. What is the score of student D?

Answer: for most of you, the score of student D would be 4 × 47 – 3 × 50 = 38.

For me, the calculation would just be 47 – 9 = 38.

Some of you might have understood what I did. Let me start explaining through a simple example. Then we shall extend our explorations to more complex problems.

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One question that will never stop coming from the students who prepare for CAT is “how to increase the vocabulary?” Suggestions such as read a lot, mug up words lists, make flash cards, subscribe to word a day etc. have always poured in from human race. Since I am the Queen of the Gadha Land, I am quite allergic to the advices that mortals always keep giving me. I like to make my own rules – Gadha Rules.

Each one of us must have followed some comic strip when we were kids but a few like me got addicted, and waste at least an hour each day even when we have left childhood at least 2 decades back. My day doesn’t complete without taking my daily dose of Archie, Peanuts, Nancy, Calvin and Hobbes, Dilbert, Garfield, Dennis etc.

But have you ever thought that these comics are a rich source of vocabulary and learning new words can really come with an incentive of laughter? Let me prove myself:

This strip is from Peanuts:

Vocabulary

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How do we find divisors of a number? For example, how do we calculate the number of divisors of 900?

Answer: 900 = 22 × 32 × 52. Therefore, any number that is a factor of 900 can have powers of 2 equal to 20, 21 or 22. Similarly, for 3, the powers can be 30, 31, or 32 and for 5 they will be 50, 51, or 52. Writing the powers in a line we have-

Divisors of a Number

Now any combination of a power of 2, a power of 3, and a power of 5 will give us a divisor. For example, in the figure, 21 × 32 × 51 will be a divisor of 900. As we can select a power of 2 in 3 ways, a power of 3 in 3 ways, and a power of 5 in 3 ways, the total number of combinations will be 3 × 3 × 3 = 27. Therefore, the number of divisors of 900 is 27.

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