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