# Similar Triangles

The current article deals with a frequently occurring topic in geometry in CAT- similarity of triangles. Most of you will notice that this topic is coming year after year in CAT and, as geometry questions in CAT are easy compared with other topics, mastering the concept is easier than mastering concepts in Algebra, Permutation, and Combination, or Probability.

# Divisibility Rules A. Divisibility by 2, 4, 8, 16, 32.

A number is divisible by 2, 4, 8, 16, 32,.. 2 when the number formed by the last one, two, three, four, five…n digits is divisible by 2, 4, 8, 16, 32,..2 respectively.

Example: 1246384 is divisible by 8 because the number formed by the last three digits i.e. 384 is divisible by 8. The number 89764 is divisible by 4 because the number formed by the last two digits, 64 is divisible by 4.

B. Divisibility by 3 and 9

A number is divisible by 3 or 9 when the sum of the digits of the number is divisible by 3 or 9 respectively.

# Remainder Theorems

It is uncanny how children pick up a lot of small habits and beliefs of their parents. Even the ones that rebel against their parents bear the subconscious resemblance to their father or mother. There is a lesson for instructors in this. It is important for them to realize that students mirror their feelings about CAT. If the instructor expresses or feels that CAT is tough, or is fearful about CAT, his students will mirror the same feeling and will be less confident. If the instructor is brazen and casual about the paper and scoffs the competition, his students would reflect the same feelings. Also, it is so necessary to have unflinching faith in one’s students. I still remember that during my school days my mother used to proudly proclaim that I was an intelligent kid. I was barely scrapping passing marks in the school exams. If truth be told I was at the bottom of the class, but my mother had her blinkers on. And because of my mother, I also believed that I was second to none. It was only years later, during my boards exams, that I took to studying seriously, and managed to outperform everyone else. I don’t know if it was my mother’s blind love for me or that she could see some bright spark in me that made her claim my intelligence but it really had great effect on my attitude. And attitude, in an exam like CAT, is everything.

Well, it’s almost everything. A keen attitude towards CAT subjects’ fundamentals also plays an important role in one’s performance.

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 digits of numbers ending in 1

What are the last two digits of 31786?

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.

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.

. 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.

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

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.