~ x+y+z+...(r terms) = n

■Non negative solutions => (n+r-1)C(r-1)

■Positive solns => (n-1)C(r-1)

~ ax+by= n

■ Non negative integral solns =

[n/lcm(a,b)]+1, if n is divisible by any of a and b.

■ If n is divisible by none, then it may either be +1 or may not be.

Means : if none is divisibel by n then ya to answer [n/lcm(a,b)]+1 hoga ya sirf [n/lcm(a,b)].

~ ● |x|+|y|= n has 4n integral solns

●|x|+|y|+|z|=n has 4n^2+2 integral solns

● |x|+|y|+|z|+|w|=n has 8n/3*(n^2+2) integral solns

~ ) x*y=N ,where N is natural number

■Number of positive solns => Number of factors of N.

■Integral soln => 2*Number of factors of N.

● Number of ways N can be expressed as product of 2 numbers => Factors/2