Question

Complete the following questions.
(a) Let S = {2, 16, 64, 512, 2048, 16384}. If four numbers are selected from
S, prove that two of them must have the product 32768.

(b) Generalize the result in part(a).

Answer 1

a)

S = { 2^1 , 2^4 , 2^6 , 2^9 , 2^11 , 2^14 }

if four number is chosen randomly we need to prove that any two of them must-have product of 32768 = 2^15

that means the sum of the power of 2 must be 15

the powers are { 1,4,6,9,11,14 }

Take any four of them we always get a sum of two = 15

let's check few of them {1,4,6,9} >>> 9+6 =15

{ 11,4,9,14,}>>11+4 = 15

likewise, there are 6C4 = 15 ways of choosing 4 numbers from 6 and all of them must have sum 15

Hence, two of them have product 32768.

b) From the above solution, we find that when 4 numbers are drawn then any two product is 2^15. here 15 is 6C4

so, by similar analogy, suppose n numbers are drawn(n<=6) then any two product must be 2^(6Cn)