GIF (not an easy one)
Algebra
Level
4
If a and b are positive integers with no common factor, What is the value of : [a/b] +[2a/b]+[3a/b]+.....+[(b-1)a/b]
(b) (a-1)/2
(b-1)(a)/2
(b-1)(a)
(b-2)(a-1)
(b-1)(a-1)/2
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1 solution
First, [ a / b ] + [ ( b − 1 ) a / b ] < = [ a ] The equality will hold iff a/b is integer, but it is not.
Therefore,
[ a / b ] + [ ( b − 1 ) a / b ] = a − 1
Similarly,
[ 2 a / b ] + [ ( b − 2 ) a / b ] = a − 1 and so on.
Now, Case 1:
b is odd.
SUM= ( a − 1 ) ∗ ( b − 1 ) / 2
Case 2:
b is even(then a is odd,because they are co-primes)
SUM= ( b − 2 ) ( a − 1 ) / 2 + [ a / 2 ] = ( b − 2 ) ( a − 1 ) / 2 + ( a − 1 ) / 2 = ( b − 1 ) ( a − 1 ) / 2