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The Official CAT 2007 Quant Thread -
30-01-2007, 12:49 PM
Hi to all,
I have tried searching all of pagalguy for a thread on Quant for 2007, but have failed and whatever threads are there are either closed or non quantitative. So i wish to start this thread, and hope it to be the official thread for CAT 2007 for quant section. Mods, please make this a sticky thread so that more threads are not opened.
I would like all quant gurus warrior, vineet and the likes to please contribute to this thread.
Lets start with the basics and fundas and get set to CRACK CAT 07.
When asked to do division in your head, it's very impressive to be able to carry the answer out to several decimal places. When dividing by a 1-digit number, it's not that difficult, either. First, you need to remember all the possible decimal equivalents for each single digit divisor. Most people will already know the first few decimal equivalents:
1/2=.5
1/3=.333...
2/3=.666...
1/4=.25
2/4=1/2=.5
3/4=.75
What are the rest? 5ths are easy, as you simply double the dividend, and place the decimal in front of that number:
1/5=.2
2/5=.4
3/5=.6
4/5=.8
With 6th, you already know 3 of the 5 decimals from above:
2/6=1/3=.333...
3/6=1/2=.5
4/6=2/3=.666...
You simply need to learn just two more 6ths:
1/6=.1666...
5/6=.8333...
7ths have a very unique pattern! Let's start with 1/7:
1/7=.142857142857142857...
All you have to do for 7ths is to remember the sequence 142857 (which repeats over and over again). Each 7th will always contain this same sequence, and only the starting point will change!
To find the appropriate starting point, take the dividend and multiply it by 14 (you should be able to do up to 14*6 in your head fairly quickly of course). Find the place in the 142857 sequence closest to this number, and you'll have the appropriate starting place!
Starting again with 1/7, we simply think (1*14=14, so 1/7 starts at the 14, and is thus equal to .142857142857...)
Here's how you figure the rest of the 7ths, with the thought process in parentheses:
With a little practice, these decimal equivalents will come to mind quickly.
Now for the full feat. To keep things simple, start working with 2 digit numbers.
Have someone choose any 2-digit number and any 1-digit number, and you can announce the result of dividing the larger number by the smaller one. For example, let's say they choose 59 divided by 6.
You should quickly realize that the closest multiple of 6 to 59, without going over, is 54 (6*9). So, the answer is 9 and 5/6ths. Instead of saying it that way, however, you remember 5/6 = 0.833, and so the answer is 9.833.
People see decimals as very complex, so this is very impressive, yet not hard to do.
For 3- and 4-digit numbers, you need to practice working through the division problem from left to right in your head.
Starting with a 3-digit example, let's try 698 divided by 7. Beginning with the leftmost digit, we quickly see that 7 won't go into 6, so we move to the next digit. 7 will go into 69 nine times, so our answer is 90-something. Taking away 63 (7*9), that leaves us with 68 to work with. 7 can go into 68 nine times, as well, so that gives us 99, with 5 as a remainder, or 99 and 5/7ths. Remember the decimal equivalent of 5/7ths? This means you can give the answer as 99.7142857 in short order.
4-digits work the same way, with one extra step, of course. 4732 divided by 6? Let's try it:
4/6=won't work
47/6=7, carrying the 5 (47-42=5), so it's 700 something
53/6=8, carrying the 5 (53-48=5), so it's 780 something
52/6=8, carrying the 4 (52-48=4), so it's 788 and 4/6, or 788 2/3
Translated into decimal form, you say "788.666".
Even if you never get comfortable with 4-digit numbers, dividing 3-digit numbers by 1-digit numbers is still impressive, especially when you can carry it out to many decimal places.
Living on a Prayer................................
The Following 132 Users Say Thank You to Krack_CAT For This Useful Post:
Re: The Official CAT 2007 Quant Thread -
30-01-2007, 02:11 PM
Quote:
Originally Posted by Krack_CAT
Can some one please throw some light on this Euler concept.
Euler's thm (a generalization of Fermat's little thm) states tht
if n is a +ve integer
a is co-prime to n
then a^{φ(n)} = 1 (mod n)
where φ(n) is Euler's totient function of n
This means tht a^{φ(n)} when divided by n gives the remainder as 1.
Finding φ(n) :
Express n in terms of its prime factors..
Say n = (x^p)(y^q)(z^r)..where x,y,z are prime nos..
then φ(n) = n * (1-{1/x}) * (1-{1/y}) * (1-{1/z})
NOTE : if n is prime, then φ(n) will always be (n-1)..After all,all the nos less than a Prime No are Co-Prime to it..
Re: The Official CAT 2007 Quant Thread -
30-01-2007, 03:12 PM
i guess the remainder when 2000^1000 is divided by 13 is: 3
here z how i did it:
2000/13 gives a remainder of (-2)
so we need to find the remainder when (-2)^1000/13
(-2)^4/13 gives a remainder of 3
so we need 2 find out the remainder of (3)^250/13
now 3^3/13 gives a remainder 1
so we break 3^250= 3^249 * 3
the 1st part wil give a remainder 1.
hence the final remainder is (1*3)=3
The Following 14 Users Say Thank You to rik_12 For This Useful Post:
Re: The Official CAT 2007 Quant Thread -
30-01-2007, 04:10 PM
Quote:
Originally Posted by Krack_CAT
Q) Find the remainder when 2000^1000 is divided by 13.
(2000^1000) mod 13 = ({154*13-2}^1000) mod 13
= ({-2}^1000) mod 13 (...coz all other terms would
involve a power of 13, which on dividin by 13 will give
rem as 0)
= ({-2}^{12*83} * {-2}^{4}) mod 13...(This is where Euler's thm Comes in..coz 12 is the euler no for 13)
= 1*({-2}^{4}) mod 13
= 16 mod 13
= 3
Re: The Official CAT 2007 Quant Thread -
30-01-2007, 04:49 PM
