Function minimum value
The function , where and are positive integers, is defined for all positive integers. If the range of contains two numbers that differ by 20, what is the least possible value of ?
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1 solution
Select two values of the function, and we have f ( y ) − f ( x ) = 2 0 . Therefore, f ( y ) − f ( x ) = a ( y ! ) + b − ( a ( x ! ) + b ) . Simplifying, we get a ( y ! − x ! ) = 2 0 . Since the function is defined for all positive integers, we try finding ordered pairs that are factors of 20. (1,20); (2,10); (4,5). the values of the first 5 factorials are shown: 1!=1; 2!=2; 3!=6; 4!=16, 5!=120. Now, try substituting values for a based off the ordered pairs. We see that if a=4, 3!-1!=5. Now we know a is 4, b can be any positive integer. But since we want to find the minimum value that f ( 1 ) can take on, we pick b=1. f ( 1 ) = 4 ( 1 ! ) + 1 = 5 .