Maths Shortcuts - 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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 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 = (2 4 * 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. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 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 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 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 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 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 . ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 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.) ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 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 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 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) . +++++++++++++++++++++++++++++++++++++++++++++++++++++++++ |x| + |y| >= |x+y| (|| stands for absolute value or modulus )...
View Full Document

This note was uploaded on 01/27/2010 for the course EE 100 taught by Professor Abc during the Spring '10 term at Punjab Engineering College.

Page1 / 8

Maths Shortcuts - 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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online