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Finding number of Factors

To find the number of factors of a given number, express the number as a product of powers of prime numbers.

In this case, 48 can be written as 16 * 3 = (24 * 3)

Now, increment the power of each of the prime numbers by 1 and multiply the result.

In this case it will be (4 + 1)*(1 + 1) = 5 * 2 = 10 (the power of 2 is 4 and the power of 3 is 1)

Therefore, there will 10 factors including 1 and 48. Excluding, these two numbers, you will have 10 – 2 = 8 factors.

Sum of n natural numbers

-> The sum of first n natural numbers = n (n+1)/2

-> The sum of squares of first n natural numbers is n (n+1)(2n+1)/6

-> The sum of first n even numbers= n (n+1)

-> The sum of first n odd numbers= n^2

Finding Squares of numbers

To find the squares of numbers near numbers of which squares are known

To find 41^2 , Add 40+41 to 1600 =1681

To find 59^2 , Subtract 60^2-(60+59) =3481

Finding number of Positive Roots

If an equation (i:e f(x)=0 ) contains all positive co-efficient of any powers of x , it has no positive roots then.

Eg: x^4+3x^2+2x+6=0 has no positive roots .

Finding number of Imaginary Roots

For an equation f(x)=0 , the maximum number of positive roots it can have is the number of sign changes in f(x) ; and the maximum number of negative roots it can have is the number of sign changes in f(-x) .

Hence the remaining are the minimum number of imaginary roots of the equation(Since we also know that the index of the maximum power of x is the number of roots of an equation.)

Reciprocal Roots

The equation whose roots are the reciprocal of the roots of the equation ax^2+bx+c is cx^2+bx+a

Roots

Roots of x^2+x+1=0 are 1,w,w^2 where 1+w+w^2=0 and w^3=1

Finding Sum of the roots

For a cubic equation ax^3+bx^2+cx+d=o sum of the roots = - b/a sum of the product of the roots taken two at a time = c/a product of the roots = -d/a

For a biquadratic equation ax^4+bx^3+cx^2+dx+e = 0 sum of the roots = - b/a sum of the product of the roots taken three at a time = c/a sum of the product of the roots taken two at a time = -d/a product of the roots = e/a

Maximum/Minimum

-> If for two numbers x+y=k(=constant), then their PRODUCT is MAXIMUM if x=y(=k/2). The maximum product is then (k^2)/4

-> If for two numbers x*y=k(=constant), then their SUM is MINIMUM if x=y(=root(k)). The minimum sum is then 2*root(k) .

Inequalties

-> x + y >= x+y ( stands for absolute value or modulus ) (Useful in solving some inequations)

-> a+b=a+b if a*b>=0 else a+b >= a+b

-> 2<= (1+1/n)^n <=3 -> (1+x)^n ~ (1+nx) if x<<<1> When you multiply each side of the inequality by -1, you have to reverse the direction of the inequality.

Product Vs HCF-LCM

Product of any two numbers = Product of their HCF and LCM . Hence product of two numbers = LCM of the numbers if they are prime to each other

AM GM HM

For any 2 numbers a>b a>AM>GM>HM>b (where AM, GM ,HM stand for arithmetic, geometric , harmonic menasa respectively) (GM)^2 = AM * HM

Sum of Exterior Angles

For any regular polygon , the sum of the exterior angles is equal to 360 degrees hence measure of any external angle is equal to 360/n. ( where n is the number of sides)

For any regular polygon , the sum of interior angles =(n-2)180 degrees

So measure of one angle in

Square-----=90

Pentagon--=108

Hexagon---=120

Heptagon--=128.5

Octagon---=135

Nonagon--=140

Decagon--=144

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Problems on clocks

Problems on clocks can be tackled as assuming two runners going round a circle , one 12 times as fast as the other . That is , the minute hand describes 6 degrees /minute the hour hand describes 1/2 degrees /minute . Thus the minute hand describes 5(1/2) degrees more than the hour hand per minute .

The hour and the minute hand meet each other after every 65(5/11) minutes after being together at midnight. (This can be derived from the above) .

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Co-ordinates

Given the coordinates (a,b) (c,d) (e,f) (g,h) of a parallelogram , the coordinates of the meeting point of the diagonals can be found out by solving for [(a+e)/2,(b+f)/2] =[ (c+g)/2 , (d+h)/2]

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Ratio

If a1/b1 = a2/b2 = a3/b3 = .............. , then each ratio is equal to (k1*a1+ k2*a2+k3*a3+..............) / (k1*b1+ k2*b2+k3*b3+..............) , which is also equal to (a1+a2+a3+............./b1+b2+b3+..........)

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Finding multiples

x^n -a^n = (x-a)(x^(n-1) + x^(n-2) + .......+ a^(n-1) ) ......Very useful for finding multiples .For example (17-14=3 will be a multiple of 17^3 - 14^3)

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Exponents

e^x = 1 + (x)/1! + (x^2)/2! + (x^3)/3! + ........to infinity 2 <>GP

-> In a GP the product of any two terms equidistant from a term is always constant .

-> The sum of an infinite GP = a/(1-r) , where a and r are resp. the first term and common ratio of the GP .

Mixtures

If Q be the volume of a vessel q qty of a mixture of water and wine be removed each time from a mixture n be the number of times this operation be done and A be the final qty of wine in the mixture then ,

A/Q = (1-q/Q)^n

Finding number of Factors

To find the number of factors of a given number, express the number as a product of powers of prime numbers.

In this case, 48 can be written as 16 * 3 = (24 * 3)

Now, increment the power of each of the prime numbers by 1 and multiply the result.

In this case it will be (4 + 1)*(1 + 1) = 5 * 2 = 10 (the power of 2 is 4 and the power of 3 is 1)

Therefore, there will 10 factors including 1 and 48. Excluding, these two numbers, you will have 10 – 2 = 8 factors.

Sum of n natural numbers

-> The sum of first n natural numbers = n (n+1)/2

-> The sum of squares of first n natural numbers is n (n+1)(2n+1)/6

-> The sum of first n even numbers= n (n+1)

-> The sum of first n odd numbers= n^2

Finding Squares of numbers

To find the squares of numbers near numbers of which squares are known

To find 41^2 , Add 40+41 to 1600 =1681

To find 59^2 , Subtract 60^2-(60+59) =3481

Finding number of Positive Roots

If an equation (i:e f(x)=0 ) contains all positive co-efficient of any powers of x , it has no positive roots then.

Eg: x^4+3x^2+2x+6=0 has no positive roots .

Finding number of Imaginary Roots

For an equation f(x)=0 , the maximum number of positive roots it can have is the number of sign changes in f(x) ; and the maximum number of negative roots it can have is the number of sign changes in f(-x) .

Hence the remaining are the minimum number of imaginary roots of the equation(Since we also know that the index of the maximum power of x is the number of roots of an equation.)

Reciprocal Roots

The equation whose roots are the reciprocal of the roots of the equation ax^2+bx+c is cx^2+bx+a

Roots

Roots of x^2+x+1=0 are 1,w,w^2 where 1+w+w^2=0 and w^3=1

Finding Sum of the roots

For a cubic equation ax^3+bx^2+cx+d=o sum of the roots = - b/a sum of the product of the roots taken two at a time = c/a product of the roots = -d/a

For a biquadratic equation ax^4+bx^3+cx^2+dx+e = 0 sum of the roots = - b/a sum of the product of the roots taken three at a time = c/a sum of the product of the roots taken two at a time = -d/a product of the roots = e/a

Maximum/Minimum

-> If for two numbers x+y=k(=constant), then their PRODUCT is MAXIMUM if x=y(=k/2). The maximum product is then (k^2)/4

-> If for two numbers x*y=k(=constant), then their SUM is MINIMUM if x=y(=root(k)). The minimum sum is then 2*root(k) .

Inequalties

-> x + y >= x+y ( stands for absolute value or modulus ) (Useful in solving some inequations)

-> a+b=a+b if a*b>=0 else a+b >= a+b

-> 2<= (1+1/n)^n <=3 -> (1+x)^n ~ (1+nx) if x<<<1> When you multiply each side of the inequality by -1, you have to reverse the direction of the inequality.

Product Vs HCF-LCM

Product of any two numbers = Product of their HCF and LCM . Hence product of two numbers = LCM of the numbers if they are prime to each other

AM GM HM

For any 2 numbers a>b a>AM>GM>HM>b (where AM, GM ,HM stand for arithmetic, geometric , harmonic menasa respectively) (GM)^2 = AM * HM

Sum of Exterior Angles

For any regular polygon , the sum of the exterior angles is equal to 360 degrees hence measure of any external angle is equal to 360/n. ( where n is the number of sides)

For any regular polygon , the sum of interior angles =(n-2)180 degrees

So measure of one angle in

Square-----=90

Pentagon--=108

Hexagon---=120

Heptagon--=128.5

Octagon---=135

Nonagon--=140

Decagon--=144

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Problems on clocks

Problems on clocks can be tackled as assuming two runners going round a circle , one 12 times as fast as the other . That is , the minute hand describes 6 degrees /minute the hour hand describes 1/2 degrees /minute . Thus the minute hand describes 5(1/2) degrees more than the hour hand per minute .

The hour and the minute hand meet each other after every 65(5/11) minutes after being together at midnight. (This can be derived from the above) .

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Co-ordinates

Given the coordinates (a,b) (c,d) (e,f) (g,h) of a parallelogram , the coordinates of the meeting point of the diagonals can be found out by solving for [(a+e)/2,(b+f)/2] =[ (c+g)/2 , (d+h)/2]

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Ratio

If a1/b1 = a2/b2 = a3/b3 = .............. , then each ratio is equal to (k1*a1+ k2*a2+k3*a3+..............) / (k1*b1+ k2*b2+k3*b3+..............) , which is also equal to (a1+a2+a3+............./b1+b2+b3+..........)

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Finding multiples

x^n -a^n = (x-a)(x^(n-1) + x^(n-2) + .......+ a^(n-1) ) ......Very useful for finding multiples .For example (17-14=3 will be a multiple of 17^3 - 14^3)

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Exponents

e^x = 1 + (x)/1! + (x^2)/2! + (x^3)/3! + ........to infinity 2 <>GP

-> In a GP the product of any two terms equidistant from a term is always constant .

-> The sum of an infinite GP = a/(1-r) , where a and r are resp. the first term and common ratio of the GP .

Mixtures

If Q be the volume of a vessel q qty of a mixture of water and wine be removed each time from a mixture n be the number of times this operation be done and A be the final qty of wine in the mixture then ,

A/Q = (1-q/Q)^n

**Some Pythagorean triplets:**

3,4,5----------(3^2=4+5)

5,12,13--------(5^2=12+13)

7,24,25--------(7^2=24+25)

8,15,17--------(8^2 / 2 = 15+17 )

9,40,41--------(9^2=40+41)

11,60,61-------(11^2=60+61)

12,35,37-------(12^2 / 2 = 35+37)

16,63,65-------(16^2 /2 = 63+65)

20,21,29-------(EXCEPTION)----------------------------------------------------------

Appolonius theorem

Appolonius theorem could be applied to the 4 triangles formed in a parallelogram.

Function

Any function of the type y=f(x)=(ax-b)/(bx-a) is always of the form x=f(y) .

Finding Squares

To find the squares of numbers from 50 to 59

For 5X^2 , use the formulae

(5X)^2 = 5^2 +X / X^2

Eg ; (55^2) = 25+5 /25

=3025

(56)^2 = 25+6/36

=3136

(59)^2 = 25+9/81

=3481

Successive Discounts

Formula for successive discounts

a+b+(ab/100)

This is used for succesive discounts types of sums.like 1999 population increses by 10% and then in 2000 by 5% so the population in 2000 now is 10+5+(50/100)=+15.5% more that was in 1999 and if there is a decrease then it will be preceeded by a -ve sign and likewise.

Rules of Logarithms:

-> loga(M)=y if and only if M=ay

-> loga(MN)=loga(M)+loga(N)

-> loga(M/N)=loga(M)-loga(N)

-> loga(Mp)=p*loga(M)

-> loga(1)=0-> loga(ap)=p

-> log(1+x) = x - (x^2)/2 + (x^3)/3 - (x^4)/4 .........to infinity [ Note the alternating sign . .Also note that the ogarithm is with respect to base e ]

3,4,5----------(3^2=4+5)

5,12,13--------(5^2=12+13)

7,24,25--------(7^2=24+25)

8,15,17--------(8^2 / 2 = 15+17 )

9,40,41--------(9^2=40+41)

11,60,61-------(11^2=60+61)

12,35,37-------(12^2 / 2 = 35+37)

16,63,65-------(16^2 /2 = 63+65)

20,21,29-------(EXCEPTION)----------------------------------------------------------

Appolonius theorem

Appolonius theorem could be applied to the 4 triangles formed in a parallelogram.

Function

Any function of the type y=f(x)=(ax-b)/(bx-a) is always of the form x=f(y) .

Finding Squares

To find the squares of numbers from 50 to 59

For 5X^2 , use the formulae

(5X)^2 = 5^2 +X / X^2

Eg ; (55^2) = 25+5 /25

=3025

(56)^2 = 25+6/36

=3136

(59)^2 = 25+9/81

=3481

Successive Discounts

Formula for successive discounts

a+b+(ab/100)

This is used for succesive discounts types of sums.like 1999 population increses by 10% and then in 2000 by 5% so the population in 2000 now is 10+5+(50/100)=+15.5% more that was in 1999 and if there is a decrease then it will be preceeded by a -ve sign and likewise.

Rules of Logarithms:

-> loga(M)=y if and only if M=ay

-> loga(MN)=loga(M)+loga(N)

-> loga(M/N)=loga(M)-loga(N)

-> loga(Mp)=p*loga(M)

-> loga(1)=0-> loga(ap)=p

-> log(1+x) = x - (x^2)/2 + (x^3)/3 - (x^4)/4 .........to infinity [ Note the alternating sign . .Also note that the ogarithm is with respect to base e ]

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