Alice writes down all 7-digit numbers using the digits 1, 2, 3, 4, 5, 6, and 7 exactly once. Prove that there are no two numbers among them where one is a multiple of the other.
Refer to
Points On A Circle.
(A) Seven points are placed on the circumference of a circle such that the distance between any two of the points, measured along the circumference, is an integer.
What is the smallest radius of the circle, given that each of the distances is unique?
(B) Seven points are placed on the circumference of a circle such that the distance between any two of the points, measured as a straight line, is an integer.
Determine the smallest radius of the circle. What is the smallest radius of the circle, given that it is rational?
Note: In Part (B) each of the distances may or may not be unique.
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