May we should check out some more important points related to numbers before jumping in to problems
Number line
http://catdumps.blogspot.com/2009/02/number-line.html
So what is LCM ?
http://catdumps.blogspot.com/2009/01/least-common-multiple-lcm.html
HCF means ....:
http://catdumps.blogspot.com/2009/01/highest-common-factor-hcf.html
Absolutes and opposites
http://catdumps.blogspot.com/2009/02/absolute-value-and-opposites.html
About Zero and one
http://catdumps.blogspot.com/2009/01/neither-prime-nor-composite.html
Face value and place value
"Just as our face never changes, similarly the face value of the number remains same. wherever the number is " .
But the place value of the number changes depending upon their position. (Eg : For the number, 6780, the place value of 7 is 700, while the face value is 7 but for the no. 67800, the place value becomes 7000).
For any two numbers
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 )
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
For any consecutive numbers:
Average of consecutive numbers=(smallest number+largest_number)/2
Number of terms in series = (largest number - samallest number + 1 )
Sum of consecutive numbers= Average X number of terms =(smallest number+largest_number) X (largest number - samallest number + 1 )/2
Reminder Theorem:
Divident=(Divisor*Quotient) + Reminder
Sum of first n Natural Numbers = n(n+1)/2
Sum of squares of first n Natural Numbers=n(n+1)(2n+1)/6
Sum of cubes of first n Natural Numbers = ( n(n+1)/2 )^2
