It is funny how one problem gives rise to another. Of course, this happens because of the irritating habit that mathematicians have of asking, â€œ if it was something else and not something like this, what would be the solution?â€ 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 wonâ€™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. Letâ€™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:
Hello World!! Now that you are well set on your journey of grammar, here’s a small lesson for you on one of the most frequent error zones in grammar- Modifiers.
Oh, by the way! Do you like coffee? It might not be a bad idea to keep a steaming mugful by your side as you read the lesson. Sip along!
Modifiers- Dangling and Misplaced:
A modifier describes, clarifies, or gives more detail about a word or a word group. A dangling modifier is a phrase that modifies the wrong word or phrase because of the absence of the word or phrase it is supposed to modify. In other words, the modifier is left “dangling”. A misplaced modifier is placed incorrectly in the sentence such that it modifies the wrong word and makes the sentence sound illogical.
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.
This kind of argumentation is known as deductive reasoning. Here, the conclusion arrived at, is a logical ‘necessity’, which you will find me referring to henceforth as an LN. The structure of the deductive argumentation is simple. We picked a set, gave it a characteristic (P1), picked an element from the set (P2), and with certainty, arrived at the conclusion that the element shall show the same characteristic.
P.S. I hope you understand that my sympathies with Ravi have nothing to do with the argument.
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.
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.