One equation three unknown
Number Theory
Level
3
Three positive integers a, b and c * are such that * a>b>c and a+ b+ c+ ab +bc +ca +abc=1000 Find a +b +c
The answer is 28.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
Adding 1 to both sides of the given equation we get the equation ( 1 + a ) ( 1 + b ) ( 1 + c ) = 1 + a + b + c + a b + a c + b c + a b c = 1 0 0 1 . Since 1 0 0 1 = 7 ∗ 1 1 ∗ 1 3 and the integers a , b , and c satisfy the inequalities a > b > c > 0 , then 1 + a = 1 3 , 1 + b = 1 1 , and 1 + c = 7 . Therefore, a + b + c = 6 + 1 0 + 1 2 = 2 8 .