Cyclicity

In CAT, MAT and other Competitive examinations like Bank PO, etc. you get questions where you need to find the last digit of numbers raised to large powers. It's almost impossible to calculate the values of such numbers manually and  hence to find digit at their unit's place.  Such problems can be solved using the concept commonly known as Cyclicity of Numbers. Here in this post I am explaining in details the concept of cyclicity and how it should be used for solving such problems.

Finding Last Digit of Any Number Raised to Any Power

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I remember back to early 2005 (before going for my post graduation), when I’d just started preparing for some competitive examination, that even though I had great ambitions, my knowledge, expertise, and confidence as a student was sorely lacking. My love for internet (as a source of information) made me stumble through various places on the web to know maths short cuts and quick maths tricks, but without much to gain from it.

Now, after having being there and facing it all as a student first and then as a facilitator (though I take good number of Quants classes for CAT aspirants, I would not liked to be called a tutor or a teacher. These titles should be reserved for people of great wisdom and learning) for over 4-5 years I felt I had a really good grip on things;  and even then, there were many things I am still struggling with.

Now with the help of QuickerMaths.com, I want to create a platform, easily accessible to the student fraternity where they can get what they are looking for; at least things related to Quantitative Aptitude, Logical Reasoning, Analytical Reasoning and Critical Reasoning - a platform where you can ask questions and where you can give answers.

Let The Process Begin

I’m starting the process by posting below an interesting formula of the topic- time and work . The aim is to initiate the process of collaborative learning. So anyone who wants to make a difference can post a comment with any explanation, formula, shortcut, trick, solved examples, etc. on the same topic (Time and Work for this post). You can also help by sharing the information of this platform with your friends or anyone who wants to help others and learn from others in the process.

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Time and Work Formula

I think the single most useful formula for the topic Time and Work is

N₁H₁D₁E₁W₂ = N₂H₂D₂E₂W₁

Where:

N₁ and N₂ = number of person

H₁ and H₂ = Hours worked by per person per day (assumed constant)

D₁ and D₂ = days

E₁ and E₂ = Efficiency

W₁ and W₂= Amount of work done

Consider this example to understand the applicability of this formula –

A piece of work can be done by 16 men in 8 days working 12 hours a day.How many men are needed to complete another work, which is three times the first one,in 24 days working 8 hours a day. The efficiency of the second group is half that of the first group?

Solution –

N₁H₁D₁E₁W₂ = N₂H₂D₂E₂W₁

16*12*8*1*3 = N₂*8*24*0.5*1

N₂ = (16*12*8*1*3)/ (8*24*0.5*1) = 48

So number of men required is 48.

Note – you can remove anything from formula is not given in the question. For example if the question would have been –

“A piece of work can be done by 16 men in 8 days working 12 hours a day.How many men are needed to complete another work, which is three times the first one,in 24 days working 8 hours a day.”

The applicable formula would have been –

N₁H₁D₁W₂ = N₂H₂D₂W₁

Since nothing is mentioned about efficiency, we remove it from both sides.