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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Numbers

It is funny how one problem gives rise to another. Of course, this happens because of the irritating habit that mathematicians have of asking, “ if it was something else and not something like this, what would be the solution?” And then you start all over again.

Here is a good problem I was discussing with a colleague-

In how many ways can you write the number 210 as a product of three integers?

210 = 2 Ã— 3 Ã— 5 Ã— 7. Since the problem asks for integers, we will first calculate for positive numbers and then assign signs to these numbers.

Your first instinct is to write down 2, 3, 5 and 7 in a row and place two partitions between them. The partitions will give you three groups of numbers. Something like this:

But you should quickly realize that in this way no matter where you place your partitions, the number 2 and 7, 3 and 7, 2 and 5 will never be together. The partition method won’t work.

In fact, the situation is similar to placing 4 similar balls in three similar boxes. How would you do that? The hard way of course:

So we have to place 4 prime numbers in 3 places. If a place remains empty after distribution that means we will assume number 1 over there. Let’s find the number of ways of distributing these prime numbers. Since the box are all same only the different grouping of number matters.

As we are talking about integers, the numbers of possible cases are:

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Calculation Techniques

The rules of business can be applied to any competitive environment. Just as businesses keep trying to achieve an extra edge and sell more and more products so do students preparing for competitive exams such as CAT 2018. The focus should always be on trying to achieve more and more. For CAT aspirants, this is the golden rule- always be focused on your goal. That means giving away all your distractions: that movie which other friends are going to watch, those phone calls, whatsapp, facebook, or twitter, those sms, those days of boredom when you do not want to study, and so much more. For every second, minute, hour or day that you lose, someone is trying to get ahead of you. The fact is that you are not going to become a CAT cracker overnight; you will have to grow inch by inch. And the other fact is that it is easier to increase your percentile to 95 or 97 from 70 or 80 percentile but it is much much tougher to increase it further to 99+. So the more you are at the top, the harder it is to grow further. And it takes focus and dedication climb day by day.

I am very fond of today’s chapter. These tricks are used by me off an on and I teach them in my first class also. The result is that all my students are able to work out calculations mentally by the end of the class. The sad fact is that they do not continue it once they get home, which is a pity. If they keep up the practise for 3 or 4 months, they would see themselves doing wonders, as I saw with myself. But practicing it day in and day out again takes focus and vision of one’s goal.

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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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averages and Alligation
Average Mixture Alligation

I recently read a student blog on TG Town questioning whether it is correct by the instructors to tell students “not to take their mock scores seriously.” After all, if a student is not performing well in mocks, something must be wrong. The question really made me ask myself the reasons for my saying so. Is it even correct to say mock scores do not matter. I think there are several reasons instructors tell students not to take their mock scores seriously. Most important of them all is motivation. Students are never ready to recognize that scoring well in a test is more a matter of temperament and question selection than that of content. Infact an average student having the art of question-picking would perform twice better than a genius student who is out to solve questions in a near about serial order. Therefore, instructors want to keep their students motivated in order to keep their enthusiasm high. The second reason for not taking mocks seriously is that the level of preparedness of students at a particular point is different. Many of the institutes start their mocks by the month of May or June and most students are not ready by then. It is inevitable that they will perform badly. If the students take it seriously, they would spend rest of their time taking more and more tests to improve themselves instead of studying and then the real harm would be done. For me, when I entered the CAT preparation, my verbal didn’t need preparation, my quant took three or four months of tweaking, but my DI took more than a year to reach a decent level. The level of preparedness for every student is different. So the question is, what do you do with your mock scores? The better thing to do is to solve your mock paper, if you can solve it on your own, you don’t have a problem with content. If you cannot, get back to studies. So don’t look hard at your scores, look hard at the paper.

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

averages and Alligation
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.

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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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