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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.
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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
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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
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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 .
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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.)
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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
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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
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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
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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) .
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x| + |y| >= |x+y| (|| stands for absolute value or modulus )
(Useful in solving some inequations)
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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
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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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If any parallelogram can be inscribed in a circle , it must be a rectangle.
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If a trapezium can be inscribed in a circle it must be an isosceles trapezium (i:e oblique sides equal).
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For an isosceles trapezium , sum of a pair of opposite sides is equal in length to the sum of the other pair of opposite sides .(i:e AB+CD = AD+BC , taken in order) .
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Area of a regular hexagon : root(3)*3/2*(side)*(side)
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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
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For three positive numbers a, b ,c
(a+b+c) * (1/a+1/b+1/c)>=9
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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)
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Appolonius theorem could be applied to the 4 triangles formed in a parallelogram.
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Area of a trapezium = 1/2 * (sum of parallel sids) * height = median * height
where median is the line joining the midpoints of the oblique sides.
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when a three digit number is reversed and the difference of these two numbers is taken , the middle number is always 9 and the sum of the other two numbers is always 9 .
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ANy function of the type y=f(x)=(ax-b)/(bx-a) is always of the form x=f(y) .
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