Quantitative Aptitude Techniques.
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Here we will discuss some more quantitative methods for typical MBA examination questions.


Highest power of a number in a factorial.
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This is a typical CAT question, which is being asked for many competitive entrance exams.
To find out the highest power of a no. in a factorial, we should find the no. of times the particular no. is repeating in that factorial. Hence we will divide the factorial no. continuously until we get a no. less than the original divisor.


Eg.Find the highest power of 5 in 100!

5)100
5)20
5)4.
Therefore the max. power of 5 in 1001! is 20+4=24

If the divisor is a composite no, we need to find the highest power of all the prime no. components and we can fix the highest power of the no, as the power of largest prime num. component.

Eg. Find the highest power of 10 in 100!
As we know 10 =5*2( both are prime numbers.)
Also 5>2.
So we need to find the highest power of 5 in 100! And this will be the answer.
(Anyway the highest power of 2 is more than that of 5, but we can’t take that since,
for getting a 10 we need both 5 and 2, so whichever is least, only that many 10s will be there).

Number of zeroes in a factorial.(n!)

The num. of zeroes will depend on the highest power of 10 in that factorial. That in turn depends on the highest power of 5( as we mentioned earlier.)

****In any number system, the number of zeroes depends on the highest power of the base number of that factorial****




Expressing a number as the difference between 2 squares.
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Suppose we have a number N=a.b
We can express N as
N= a.b=(a+b/2)^2-(a-b/2)^2
So we conclude that
In order for N to be an integer, both a &b should be either odd or even.
****So any multiple of 4 can be expressed as the difference between 2 squares.****

Lets take an example.

a.How many ways we can express 24 as the difference between 2 squares?
Explanation:-

24=1*24(format N=a*b)
24=2*12
24=3*8
24=4*6

Here 2*12 and 4*6 are the 2 forms where both a and b are even or odd. So 24 can be expressed as the difference between 2 squares in 2 ways.

2*12 as 7^2-5^2
4*6 as 5^2-1^2

Odd
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All odd prime no. can be expressed in only 1 way
Odd composite no. can be expressed in more than 1 way

Even
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4 can be expressed in only 1 way.
Prime multiple of 4 can be expressed in only 1 form.
Other multiples of 4 can be expressed in more than 1 form.
Non multiples of 4 can’t be expressed as difference between 2 squares.

Eg.
How many no. are there below 1000,that can’t be expressed as a difference between two squares?
Explanation:-

All odd num. can be expressed as the difference between two squares.
In even num., only which are non-multiples of 4 can’t be expressed.
So in 999 num. 499 even numbers are there.
Out of which ,250 are multiples of 4.
So non- multiples are 499-250=249.
So the answer is 249.